From 9c0f622e0ce4990ee0cb404b98bdda8d4a5ea467 Mon Sep 17 00:00:00 2001 From: team3 Date: Thu, 18 Jun 2026 14:30:21 +0200 Subject: [PATCH] update --- backend/bausteine.py | 186 +- backend/config.py | 4 + backend/elements.py | 2 +- backend/guide.py | 40 +- backend/models.py | 32 + backend/pipeline.py | 26 + backend/regeln.py | 7 +- backend/routes.py | 50 +- frontend/src/App.vue | 78 +- frontend/src/api.js | 24 + .../src/components/BausteineUebersicht.vue | 211 + frontend/src/components/TopicSidebar.vue | 414 +- templates/Prompt/Relevanz-Mapping.md | 18 + templates/Prompt/Relevanz-Recherche.md | 19 + uni/aak (Copy)/Skript.pdf | 4456 +++++++++ uni/aak (Copy)/Skript.txt | 2562 ++++++ uni/aak (Copy)/alle_hausaufgaben.pdf | 7959 +++++++++++++++++ uni/aak (Copy)/alle_hausaufgaben.txt | 817 ++ uni/aak (Copy)/alle_klausuren.pdf | 5320 +++++++++++ uni/aak (Copy)/alle_klausuren.txt | 863 ++ uni/aak (Copy)/alle_praesenzaufgaben.pdf | Bin 0 -> 648498 bytes uni/aak (Copy)/alle_praesenzaufgaben.txt | 480 + uni/aak/Skript.pdf | 4456 +++++++++ uni/aak/Skript.txt | 2562 ++++++ uni/aak/alle_hausaufgaben.pdf | 7959 +++++++++++++++++ uni/aak/alle_hausaufgaben.txt | 817 ++ uni/aak/alle_klausuren.pdf | 5320 +++++++++++ uni/aak/alle_klausuren.txt | 863 ++ uni/aak/alle_praesenzaufgaben.pdf | Bin 0 -> 648498 bytes uni/aak/alle_praesenzaufgaben.txt | 480 + 30 files changed, 45808 insertions(+), 217 deletions(-) create mode 100644 frontend/src/components/BausteineUebersicht.vue create mode 100644 templates/Prompt/Relevanz-Mapping.md create mode 100644 templates/Prompt/Relevanz-Recherche.md create mode 100644 uni/aak (Copy)/Skript.pdf create mode 100644 uni/aak (Copy)/Skript.txt create mode 100644 uni/aak (Copy)/alle_hausaufgaben.pdf create mode 100644 uni/aak (Copy)/alle_hausaufgaben.txt create mode 100644 uni/aak (Copy)/alle_klausuren.pdf create mode 100644 uni/aak (Copy)/alle_klausuren.txt create mode 100644 uni/aak (Copy)/alle_praesenzaufgaben.pdf create mode 100644 uni/aak (Copy)/alle_praesenzaufgaben.txt create mode 100644 uni/aak/Skript.pdf create mode 100644 uni/aak/Skript.txt create mode 100644 uni/aak/alle_hausaufgaben.pdf create mode 100644 uni/aak/alle_hausaufgaben.txt create mode 100644 uni/aak/alle_klausuren.pdf create mode 100644 uni/aak/alle_klausuren.txt create mode 100644 uni/aak/alle_praesenzaufgaben.pdf create mode 100644 uni/aak/alle_praesenzaufgaben.txt diff --git a/backend/bausteine.py b/backend/bausteine.py index 002d7d1..99e8136 100644 --- a/backend/bausteine.py +++ b/backend/bausteine.py @@ -24,11 +24,11 @@ from jsonio import read_json_file as _json_datei from paths import arbeit_dir, bausteine_path, project_dir, subbausteine_path, quelle_path, quelle_crawl_dir, safe_ordner from crawl import crawl from pipeline import ( - CANCELLED, FAILED, GenContext, _extra, _log, _prompt, _race, _rest_schema, + CANCELLED, FAILED, GenContext, _extra, _log, _prompt, _race, _relevanz_schema, _rest_schema, _runde_schema, _semaphore, _str_liste, _stufen_schema, _timeout, run_single_slot, ) from textkit import ( - _eindeutige_titel, _norm_titel, _parse_auswahl, _parse_subbausteine, _titel, + _eindeutige_titel, _lade_bausteine, _norm_titel, _parse_auswahl, _parse_subbausteine, _titel, _titel_aufloesen, _titel_index, _vormerge, ) @@ -50,6 +50,7 @@ BAUSTEINE_STEPS = ( "Recherche", "Konsolidierung", "Klärung", "Subbausteine finden", "Subbausteine wählen", "Subbausteine klären", "Stufen finden", "Stufen wählen", "Stufen klären", + "Relevanz finden", "Relevanz wählen", "Relevanz klären", ) @@ -78,6 +79,36 @@ def _crawl_fertig(topic: str) -> bool: return (quelle_crawl_dir(topic) / ".done").exists() # Marker erst bei sauberem Abschluss +_STUFEN = ("einfach", "mittel", "schwer") + + +def lade_uebersicht(topic: str) -> list[dict]: + """Strukturierte Baustein-Liste für die Übersicht: Titel + Beschreibung + Subbausteine/Stufen. + + Verbindet bausteine.md (Nummer/Titel/Beschreibung) mit der Sidecar subbausteine.json + (Key = Titel). Fehlt die Sidecar, sind die Subbaustein-Listen leer. + """ + entries = _lade_bausteine(_read(bausteine_path(topic))) + sidecar = _json_datei(subbausteine_path(topic)) + sidecar = sidecar if isinstance(sidecar, dict) else {} + out = [] + for num, entry in entries.items(): + titel = _titel(entry) + teile = entry.split(" — ", 1) + beschreibung = teile[1].strip() if len(teile) == 2 else "" + subbausteine = [ + { + "titel": t, + "stufe": s.get("stufe") if s.get("stufe") in _STUFEN else "mittel", + "relevanz": s.get("relevanz") if s.get("relevanz") in ("relevant", "rand") else None, + } + for s in (sidecar.get(titel) or []) + if isinstance(s, dict) and (t := str(s.get("titel", "")).strip()) + ] + out.append({"num": num, "titel": titel, "beschreibung": beschreibung, "subbausteine": subbausteine}) + return out + + def _bausteine_steps(topic: str) -> tuple: """Schritte je Quelle: link bekommt vorne „Quelle laden", projekt zusätzlich „Ergänzung". @@ -89,6 +120,7 @@ def _bausteine_steps(topic: str) -> tuple: rest = ( "Subbausteine finden", "Subbausteine wählen", "Subbausteine klären", "Stufen finden", "Stufen wählen", "Stufen klären", + "Relevanz finden", "Relevanz wählen", "Relevanz klären", ) mitte = base + (("Ergänzung",) if q["type"] == "projekt" else ()) + rest return (("Quelle laden",) if q["type"] == "link" else ()) + mitte @@ -117,7 +149,7 @@ def _bausteine_files(topic: str) -> dict: def _alle_slot_dateien(files: dict) -> list[Path]: arbeit = files["arbeit"] # Subbaustein-/Stufen-Slots sind pro Chunk dynamisch — per Glob einsammeln. - dyn = (list(arbeit.glob("subbaustein-*")) + list(arbeit.glob("stufe-*"))) if arbeit.is_dir() else [] + dyn = (list(arbeit.glob("subbaustein-*")) + list(arbeit.glob("stufe-*")) + list(arbeit.glob("relevanz-*"))) if arbeit.is_dir() else [] return [ *files["recherche"], files["recherche_mapping"], *(p for slots in files["auswahl"].values() for p in slots), @@ -153,8 +185,12 @@ def _resume_step(topic: str) -> int: return _step_idx(topic, "Klärung") if q["type"] == "projekt" and not files["ergaenzung"].exists(): return _step_idx(topic, "Ergänzung") - if _sidecar_schema(_json_datei(files["sidecar"])) is not None: - return len(_bausteine_steps(topic)) + sidecar = _json_datei(files["sidecar"]) + if _sidecar_schema(sidecar) is not None: + # Stufen fertig; nur noch Relevanz offen? + if _relevanz_komplett(sidecar): + return len(_bausteine_steps(topic)) + return _step_idx(topic, "Relevanz finden") if _sub_roh_schema(_json_datei(files["sub_roh"])) is None: return _step_idx(topic, "Subbausteine finden") return _step_idx(topic, "Stufen finden") @@ -287,7 +323,7 @@ def _sub_roh_schema(data): def _sidecar_schema(data): - """{Baustein-Titel: [{titel, stufe}, …]} → dict · sonst None (finale Sidecar).""" + """{Baustein-Titel: [{titel, stufe}, …]} → dict · sonst None (Sidecar mit Stufen).""" if not isinstance(data, dict) or not data: return None for v in data.values(): @@ -299,6 +335,17 @@ def _sidecar_schema(data): return data +def _relevanz_komplett(data) -> bool: + """Jeder Subbaustein der Sidecar trägt eine gültige Relevanz (relevant/rand)?""" + if not isinstance(data, dict) or not data: + return False + return all( + isinstance(s, dict) and s.get("relevanz") in ("relevant", "rand") + for v in data.values() if isinstance(v, list) + for s in v + ) + + def _read(p: Path) -> str: return p.read_text(encoding="utf-8") if p.exists() else "" @@ -574,6 +621,115 @@ async def _stufen_block(ctx: GenContext, set_p, files: dict, roh: dict, instruct return sidecar +async def _relevanz_block(ctx: GenContext, set_p, files: dict, sidecar: dict, instructions: str) -> dict | None: + """Block D: drei Phasen mit Barriere — Finden (relevant/rand), Wählen (Vote), Klären. + Items aus der Sidecar; lokale IDs 1..n pro Paket → globale gid. + → {gid: relevanz} oder None bei Abbruch/Recherche-Fehler. Default bei Lücke/Streit: 'relevant'.""" + topic, provider, is_cancelled = ctx.topic, ctx.provider, ctx.is_cancelled + arbeit = files["arbeit"] + items = [(titel, sub["titel"]) for titel, subs in sidecar.items() for sub in subs] # globale id = index+1 + if not items: + return {} + chunks = _chunk_nums(list(range(len(items))), _n_chunks(len(items), STUFE_CHUNK)) + n = len(chunks) + + def rater_paths(c): + return [arbeit / f"relevanz-c{c}-{i}.json" for i in (1, 2, 3)] + + def lset(item_idxs): + return set(range(1, len(item_idxs) + 1)) + + # Phase „Relevanz finden": pro Paket 3 Rater (min. 2), lokale IDs. + async def _rate(c, item_idxs): + local_set = lset(item_idxs) + paths = rater_paths(c) + vorhanden = sum(1 for p in paths if _relevanz_schema(_json_datei(p), local_set)) + if vorhanden >= 2: + return True + enum = "\n".join(f"{k}. [{items[j][0]}] {items[j][1]}" for k, j in enumerate(item_idxs, 1)) + offen = [(i, p) for i, p in enumerate(paths, 1) if not _relevanz_schema(_json_datei(p), local_set)] + slots = [{ + "key": f"bausteine-{topic}-relevanz-c{c}-{i}", + "prompt": _prompt("Relevanz-Recherche", topic=topic, subbausteine=enum, out_path=p, extra=_extra(instructions)), + "role": "fast", "capabilities": "files", + "payload": (lambda result, p=p, ids=local_set: _relevanz_schema(_json_datei(p), ids)), + } for i, p in offen] + neu = await _race(topic, f"Relevanz Paket {c}", slots, 2 - vorhanden, _timeout("relevanz", len(item_idxs)), provider, cancelled=is_cancelled, grace=KONSENS_GRACE) + return not is_cancelled() and neu is not None + + set_p(f"Relevanz finden ({n} Pakete)…", step=_step_idx(topic, "Relevanz finden")) + oks = await asyncio.gather(*[_rate(c, idxs) for c, idxs in enumerate(chunks, 1)], return_exceptions=True) + if is_cancelled(): + return None + if not all(ok is True for ok in oks): + _bausteine_errors[topic] = "Relevanz fehlgeschlagen (Recherche)" + return None + + # Phase „Relevanz wählen": Code-Vote je Paket → (ergebnis, strittig). + set_p(f"Relevanz wählen ({n} Pakete)…", step=_step_idx(topic, "Relevanz wählen")) + vote_by_c = {} + for c, item_idxs in enumerate(chunks, 1): + local_set = lset(item_idxs) + rater = [d for p in rater_paths(c) if (d := _relevanz_schema(_json_datei(p), local_set))] + ergebnis: dict[int, str] = {} + strittig: dict[int, list[str]] = {} + for k in range(1, len(item_idxs) + 1): + stimmen = [d[k] for d in rater if k in d] + zaehler: dict[str, int] = {} + for s in stimmen: + zaehler[s] = zaehler.get(s, 0) + 1 + best = max(zaehler.values(), default=0) + gewinner = [s for s, v in zaehler.items() if v == best] + if len(gewinner) == 1 and best >= 2: + ergebnis[k] = gewinner[0] + else: + strittig[k] = stimmen + vote_by_c[c] = (ergebnis, strittig) + + # Phase „Relevanz klären": Judge je Paket mit Strittigem, alle parallel. + async def _klaere(c, item_idxs): + ergebnis, strittig = vote_by_c[c] + if strittig: + judge_path = arbeit / f"relevanz-final-c{c}.json" + entsch = _relevanz_schema(_json_datei(judge_path), set(strittig)) + if entsch is None: + strittig_block = "\n".join( + f"{k}. [{items[item_idxs[k - 1]][0]}] {items[item_idxs[k - 1]][1]} — Stimmen: {', '.join(stimmen) or 'keine'}" + for k, stimmen in strittig.items() + ) + status, entsch = await run_single_slot( + ctx, f"Relevanz-Klärung {c}", + key=f"bausteine-{topic}-relevanz-final-c{c}", + prompt=_prompt("Relevanz-Mapping", topic=topic, strittig=strittig_block, out_path=judge_path, extra=_extra(instructions)), + role="judge", capabilities="files", + payload=lambda result, p=judge_path, ids=set(strittig): _relevanz_schema(_json_datei(p), ids), + timeout=_timeout("relevanz_check", len(strittig)), + ) + if status == FAILED: + _log(topic, f"Relevanz-Klärung Paket {c} fehlgeschlagen — Default 'relevant'") + entsch = entsch if isinstance(entsch, dict) else {} + # Strittige ohne Entscheid → 'relevant' (nie versehentlich ausschließen). + ergebnis = {**{k: "relevant" for k in strittig}, **ergebnis, **entsch} + return {item_idxs[k - 1] + 1: rel for k, rel in ergebnis.items()} + + set_p(f"Relevanz klären ({n} Pakete)…", step=_step_idx(topic, "Relevanz klären")) + parts = await asyncio.gather(*[_klaere(c, idxs) for c, idxs in enumerate(chunks, 1)], return_exceptions=True) + if is_cancelled(): + return None + relevanz_by_id: dict[int, str] = {} + for c, part in enumerate(parts, 1): + if not isinstance(part, dict): + # Klärung ist nicht fatal: Vote-Ergebnis + Default 'relevant' für Strittige. + if isinstance(part, BaseException): + _log(topic, f"Relevanz-Klärung Paket {c}: {type(part).__name__}: {part}") + ergebnis, strittig = vote_by_c[c] + item_idxs = chunks[c - 1] + merged = {**{k: "relevant" for k in strittig}, **ergebnis} + part = {item_idxs[k - 1] + 1: s for k, s in merged.items()} + relevanz_by_id.update(part) + return relevanz_by_id + + async def generate_bausteine(topic: str, instructions: str = "", provider: str = DEFAULT_PROVIDER) -> None: if topic in _bausteine_progress: return @@ -834,6 +990,24 @@ async def generate_bausteine(topic: str, instructions: str = "", provider: str = if sidecar is None: return atomic_write_json(files["sidecar"], sidecar, indent=1) + + # Block D: Relevanz je Subbaustein (relevant/rand) → in die Sidecar mergen. + # Eigene Phase nach den Stufen; treibt das ProGuide-Format (alle Bausteine + # mit ≥1 relevantem Subbaustein) und filtert Rand-Subs aus den Guides. + sidecar = _json_datei(files["sidecar"]) + if _sidecar_schema(sidecar) is not None and not _relevanz_komplett(sidecar): + relevanz_by_id = await _relevanz_block(ctx, set_p, files, sidecar, instructions) + if is_cancelled(): + abgebrochen() + return + if relevanz_by_id is None: + return # Fehler ist gesetzt + gid = 0 + for subs in sidecar.values(): + for sub in subs: + gid += 1 + sub["relevanz"] = relevanz_by_id.get(gid, "relevant") + atomic_write_json(files["sidecar"], sidecar, indent=1) except Exception as e: log.exception("[%s] Bausteine-Generierung fehlgeschlagen", topic) _bausteine_errors[topic] = str(e)[:2000] diff --git a/backend/config.py b/backend/config.py index 30d6813..6771aea 100644 --- a/backend/config.py +++ b/backend/config.py @@ -6,6 +6,7 @@ STORAGE_DIR = PROJECT_ROOT / "storage" FRONTEND_DIST = PROJECT_ROOT / "frontend" / "dist" DB_PATH = STORAGE_DIR / "creator.db" PROJECTS_DIR = PROJECT_ROOT / "projects" +UNI_DIR = PROJECT_ROOT / "uni" MAX_CONCURRENT_GENERATIONS = 10 @@ -42,6 +43,8 @@ TIMEOUTS = { "subbaustein_check": (300, 15), # Judge entscheidet strittige Subbausteine im Chunk "stufe": (300, 10), # Subbausteine einstufen je Chunk "stufe_check": (300, 10), # Judge entscheidet strittige Stufen im Chunk + "relevanz": (300, 10), # Subbausteine relevant/rand je Chunk + "relevanz_check": (300, 10), # Judge entscheidet strittige Relevanz im Chunk "writer": (600, 120), # pro Section im Chunk "lese_check": (300, 10), # pro Section im Paket "onepager_recherche": (900, 0), @@ -55,6 +58,7 @@ TIMEOUTS = { FORMAT_ANTEIL = { "MiniGuide": (0.05, 0.10, 8, "einen kompakten Anfänger-Guide — der schnelle Einstieg ins Thema"), "Guide": (0.25, 0.35, 20, "einen ausführlichen Anfänger-Guide — ein solides Fundament im Thema"), + "ProGuide": (0.0, 1.0, 0, "einen fokussierten Guide — alles Relevante ohne Randthemen"), "FullGuide": (0.90, 1.00, 0, "einen Komplett-Guide — das ganze Thema"), } diff --git a/backend/elements.py b/backend/elements.py index 9369c79..577033e 100644 --- a/backend/elements.py +++ b/backend/elements.py @@ -71,7 +71,7 @@ def _topic_context(topic: str, limit: int = 12000) -> str: bp = bausteine_path(topic) if bp.exists(): parts.append(bp.read_text(encoding="utf-8")) - for fmt in ("FullGuide", "Guide", "MiniGuide", "OnePager"): + for fmt in ("FullGuide", "ProGuide", "Guide", "MiniGuide", "OnePager"): content = _json_datei(guide_content_path(topic, fmt)) if content: for ch in content.get("chapters", []): diff --git a/backend/guide.py b/backend/guide.py index 4add19a..99c4681 100644 --- a/backend/guide.py +++ b/backend/guide.py @@ -407,13 +407,26 @@ async def _generate_sections( "Wähle, was diesem Zweck dient — lass weg, was dafür nicht nötig ist." ) - # Schritt 0: Auswahl-Konsens (5 Agenten → Voting → Mapping → Klärungs-Loop) - auswahl = await _konsens_auswahl( - ctx, files, entries, k_min, k_max, auswahl_auftrag, format_name, - bausteine_liste, instructions, - ) - if auswahl is None: - return None + # Subbausteine je Baustein (Sidecar) — früh geladen: ProGuide wählt darüber aus, + # Nicht-FullGuide filtert später Rand-Subbausteine. Fehlt sie → {} (Fallback). + subs_raw = _load_subbausteine(topic) + + # Schritt 0: Auswahl. ProGuide ist deterministisch — alle Bausteine mit ≥1 + # relevantem Subbaustein. Fallback (keine Relevanz-Daten): alle Bausteine. + if format_name == "ProGuide": + auswahl = [num for num in entries + if any(s.get("relevanz") == "relevant" for s in subs_raw.get(_titel(entries[num]), []))] + if not auswahl: + _log(topic, "ProGuide: keine Relevanz-Daten — alle Bausteine genommen") + auswahl = list(entries) + else: + # Auswahl-Konsens (5 Agenten → Voting → Mapping → Klärungs-Loop) + auswahl = await _konsens_auswahl( + ctx, files, entries, k_min, k_max, auswahl_auftrag, format_name, + bausteine_liste, instructions, + ) + if auswahl is None: + return None sel_entries = {num: entries[num] for num in auswahl} soll = len(sel_entries) @@ -487,9 +500,16 @@ async def _generate_sections( # dieselbe Aufteilung: 1 Agent je ~30 Bausteine (gedeckelt). total_sections = sum(len(c["nums"]) for c in plan) chunks = _split_chunks(plan, min(WRITER_MAX, max(1, math.ceil(total_sections / WRITER_SECTIONS)))) - # Subbausteine je Baustein (Sidecar). Fehlt sie → leer → Inhalte/Writer laufen - # auf Baustein-Granularität weiter (Fallback für Alt-Themen ohne Sidecar). - subs_by_titel = _load_subbausteine(topic) + # Subbausteine je Baustein. Rand-Subbausteine fliegen aus allen Guides AUSSER + # FullGuide (Fokus). Subs ohne Relevanz-Feld (Alt-Themen) bleiben erhalten. + # Fehlt die Sidecar → leer → Inhalte/Writer laufen auf Baustein-Granularität. + subs_by_titel = subs_raw + if format_name != "FullGuide": + subs_by_titel = { + t: [s for s in subs if s.get("relevanz") != "rand"] + for t, subs in subs_raw.items() + } + subs_by_titel = {t: subs for t, subs in subs_by_titel.items() if subs} zuteilungen = [_zuteilung_subs(chunk, entries, subs_by_titel) for chunk in chunks] chunk_sizes = [sum(len(c["nums"]) for c in chunk) for chunk in chunks] writer_count = len(chunks) diff --git a/backend/models.py b/backend/models.py index b2ebe65..f889b8c 100644 --- a/backend/models.py +++ b/backend/models.py @@ -5,6 +5,7 @@ FormatType = Literal[ "OnePager", "MiniGuide", "Guide", + "ProGuide", "FullGuide", ] @@ -50,6 +51,37 @@ class ProjectResponse(BaseModel): name: str +class FolderResponse(BaseModel): + name: str + ort: str # relativer Pfad ab Repo-Root (z.B. "projects/foo") + + +class BausteineQuelleUpdate(BaseModel): + topic: str = Field(min_length=1, max_length=100) + type: SourceType = "thema" + ort: str = Field(default="", max_length=2000) + spec: str = Field(default="", max_length=2000) + + +class BausteineQuelleResponse(BaseModel): + type: SourceType + ort: str + spec: str + + +class SubbausteinInfo(BaseModel): + titel: str + stufe: Literal["einfach", "mittel", "schwer"] + relevanz: Literal["relevant", "rand"] | None = None + + +class BausteinUebersicht(BaseModel): + num: int + titel: str + beschreibung: str = "" + subbausteine: list[SubbausteinInfo] = [] + + class ProviderInfo(BaseModel): id: str available: bool diff --git a/backend/pipeline.py b/backend/pipeline.py index d3bc89b..36e48d3 100644 --- a/backend/pipeline.py +++ b/backend/pipeline.py @@ -158,6 +158,32 @@ def _stufen_schema(data, ids: set[int] | None = None): return out +_RELEVANZ = ("relevant", "rand") + + +def _relevanz_schema(data, ids: set[int] | None = None): + """{"relevanz": {"1": "relevant", …}} → {id: relevanz} · sonst None. + + Relevanz ∈ {relevant, rand} (binär). Wie `_stufen_schema`: sind `ids` gegeben, + müssen mindestens diese abgedeckt sein (Extras erlaubt). + """ + if not isinstance(data, dict) or not isinstance(data.get("relevanz"), dict) or not data["relevanz"]: + return None + out: dict[int, str] = {} + for k, v in data["relevanz"].items(): + try: + num = int(k) + except (ValueError, TypeError): + return None + wert = str(v).strip().casefold() + if wert not in _RELEVANZ: + return None + out[num] = wert + if ids is not None and not ids <= set(out): + return None + return out + + _MAX_RESTARTS = 2 diff --git a/backend/regeln.py b/backend/regeln.py index d4fcefa..95a179d 100644 --- a/backend/regeln.py +++ b/backend/regeln.py @@ -18,10 +18,11 @@ from paths import bausteine_path, guide_content_path from textkit import _norm_titel MAX_OFFENE_GUIDES = 3 -VORSTUFE = {"Guide": "MiniGuide", "FullGuide": "Guide"} +VORSTUFE = {"Guide": "MiniGuide", "ProGuide": "Guide", "FullGuide": "Guide"} # Welches Niveau die Vorstufe erreichen muss, um das Format freizuschalten. -FREISCHALT_LEVEL = {"Guide": "absolviert", "FullGuide": "verstanden"} -FORMATE = ("MiniGuide", "Guide", "FullGuide") +# ProGuide hängt parallel zu FullGuide an „Guide absolviert" — kein Pflicht-Gate vor FullGuide. +FREISCHALT_LEVEL = {"Guide": "absolviert", "ProGuide": "absolviert", "FullGuide": "verstanden"} +FORMATE = ("MiniGuide", "Guide", "ProGuide", "FullGuide") async def lade_lernstand() -> tuple[list[dict], dict[str, set[str]], dict[str, dict[str, set[str]]]]: diff --git a/backend/routes.py b/backend/routes.py index 0fb4f6e..0b0446e 100644 --- a/backend/routes.py +++ b/backend/routes.py @@ -7,7 +7,7 @@ from fastapi import APIRouter, HTTPException from fastapi.responses import FileResponse from agents import provider_available -from config import PROJECTS_DIR, PROVIDERS +from config import PROJECTS_DIR, UNI_DIR, PROVIDERS from database import ( create_guide, delete_guide, get_guide, list_guides, create_topic, list_topics as db_list_topics, delete_topic, @@ -16,7 +16,7 @@ from database import ( get_vertiefung, set_vertiefung, list_vertiefungen, list_baustein_progress, set_baustein_score, set_baustein_streak, set_baustein_absolviert, set_baustein_verstanden, set_baustein_gemeistert, delete_baustein_daten, ) -from bausteine import generate_bausteine, cancel_bausteine, bausteine_status, active_bausteine, reset_bausteine +from bausteine import generate_bausteine, cancel_bausteine, bausteine_status, active_bausteine, reset_bausteine, lade_quelle, lade_uebersicht from elements import generate_element, chat_with_guide, chat_with_element, check_element, style_element, refine_suggestion from lernen import NOETIG, MASTERY, MEISTERN, baustein_chat, baustein_diskussion, baustein_element_anlegen, pruefung_bewertung, pruefung_bewertung_schnell, pruefung_frage, score_berechnen, streak_berechnen, vertiefung_generieren from guide import generate_guide, guide_slot_dateien @@ -31,6 +31,7 @@ from models import ( ElementUpdateRequest, ElementCheckRequest, ElementCheckResponse, ElementStyleResponse, ElementRefineRequest, ElementRefineResponse, ProgressUpdate, ProgressResponse, ProjectResponse, ProviderInfo, + FolderResponse, BausteineQuelleUpdate, BausteineQuelleResponse, BausteinUebersicht, VertiefungRequest, VertiefungResponse, BausteinChatRequest, BausteinChatResponse, BausteinPruefungRequest, BausteinPruefungResponse, BausteinLernstandResponse, @@ -113,6 +114,18 @@ async def remove_project(name: str): return {"ok": True} +@router.get("/folders", response_model=list[FolderResponse]) +async def list_folders(kind: str): + """Ordner für die Quellen-Auswahl: kind=projekt → projects/, kind=uni → uni/.""" + base = {"projekt": (PROJECTS_DIR, "projects"), "uni": (UNI_DIR, "uni")}.get(kind) + if base is None: + raise HTTPException(400, "kind muss 'projekt' oder 'uni' sein") + root, prefix = base + if not root.is_dir(): + return [] + return [{"name": e.name, "ort": f"{prefix}/{e.name}"} for e in sorted(root.iterdir()) if e.is_dir()] + + # --- Bausteine --- @router.get("/bausteine/status", response_model=BausteineStatusResponse) @@ -161,6 +174,39 @@ async def remove_bausteine(topic: str): return {"ok": True} +def _validate_quelle(typ: str, ort: str) -> None: + """Quellen-Eingabe prüfen (gleiche Regeln wie beim Erstellen).""" + if typ in ("projekt", "uni"): + ordner = safe_ordner(ort) + if ordner is None or not ordner.is_dir(): + raise HTTPException(400, "Ordner ungültig oder nicht gefunden (Pfad relativ zum Projekt-Root, kein ../).") + elif typ == "link": + if not ort.lower().startswith(("http://", "https://")): + raise HTTPException(400, "Link muss mit http:// oder https:// beginnen.") + + +@router.get("/bausteine/quelle", response_model=BausteineQuelleResponse) +async def get_bausteine_quelle(topic: str): + return lade_quelle(topic) + + +@router.put("/bausteine/quelle", response_model=BausteineQuelleResponse) +async def update_bausteine_quelle(req: BausteineQuelleUpdate): + """Nur speichern — KEINE Neugenerierung. Quellen-/Spec-Wahl überschreiben.""" + topic, typ, ort = req.topic.strip(), req.type, req.ort.strip() + _validate_quelle(typ, ort) + qp = quelle_path(topic) + qp.parent.mkdir(parents=True, exist_ok=True) + daten = {"type": typ, "ort": ort, "spec": req.spec.strip()} + atomic_write_json(qp, daten) + return daten + + +@router.get("/bausteine/uebersicht", response_model=list[BausteinUebersicht]) +async def get_bausteine_uebersicht(topic: str): + return lade_uebersicht(topic) + + # --- Baustein-Lernen: Vertiefung, Chat, Prüfung --- async def _bester_text(topic: str, baustein: str) -> str | None: diff --git a/frontend/src/App.vue b/frontend/src/App.vue index 5a3585d..dc287bf 100644 --- a/frontend/src/App.vue +++ b/frontend/src/App.vue @@ -1,14 +1,14 @@ + + + + diff --git a/frontend/src/components/TopicSidebar.vue b/frontend/src/components/TopicSidebar.vue index 39b9868..5643f88 100644 --- a/frontend/src/components/TopicSidebar.vue +++ b/frontend/src/components/TopicSidebar.vue @@ -1,10 +1,10 @@ @@ -242,7 +288,7 @@ function confirmDeleteProject(name) {
- +
@@ -254,12 +300,13 @@ function confirmDeleteProject(name) { class="dlg-input" v-model="form.sourceOrt" placeholder="https://…" @keyup.enter="createThema" /> - -

{{ sourceHint }}

+ > + + +
@@ -283,66 +330,60 @@ function confirmDeleteProject(name) {
{{ line }}
-
-
- Bausteine - Pausiert - +
+
+ +
-
- @@ -582,59 +636,35 @@ function confirmDeleteProject(name) { } .topic-list li { - padding: 0.6rem 1rem; - cursor: pointer; font-size: 0.9rem; color: var(--text); - transition: background 0.15s; - display: flex; - justify-content: space-between; - align-items: center; } -.topic-list li:hover { +.topic-row { + display: flex; + align-items: center; + justify-content: space-between; + gap: 6px; + padding: 0.6rem 1rem; + cursor: pointer; + transition: background 0.15s; +} + +.topic-row:hover { background: var(--accent-soft); } -.topic-list li.active { +.topic-list li.active .topic-row { background: var(--accent-soft); color: var(--accent-hover); font-weight: 600; } -.delete-topic { - display: none; - background: none; - border: none; - color: var(--danger); - font-size: 1.1rem; - cursor: pointer; - padding: 0 2px; - line-height: 1; -} - -.topic-list li:hover .delete-topic { - display: block; -} - -.projects-divider { - cursor: default; - font-size: 0.7rem; - text-transform: uppercase; - letter-spacing: 0.05em; - color: var(--text-faint); - font-weight: 700; - padding: 0.6rem 1rem 0.3rem; - margin-top: 0.4rem; - border-top: 1px solid var(--border); -} - -.projects-divider:hover { - background: none; -} - -.topic-list li.project-item span::before { - content: '📁 '; +.topic-name { + flex: 1; + overflow: hidden; + text-overflow: ellipsis; + white-space: nowrap; } /* Format section */ @@ -642,7 +672,7 @@ function confirmDeleteProject(name) { display: flex; align-items: center; gap: 8px; - cursor: default; + cursor: pointer; } .step-dots { @@ -756,13 +786,14 @@ function confirmDeleteProject(name) { .format-name { flex: 1; + min-width: 0; background: none; border: none; text-align: left; font-size: 0.85rem; padding: 4px 8px; border-radius: 4px; - cursor: default; + cursor: pointer; color: var(--text-faint); display: flex; align-items: center; @@ -770,6 +801,55 @@ function confirmDeleteProject(name) { gap: 8px; } +/* Accordion: Pfeil-Toggle + aufklappendes Aktions-Panel */ +.panel-toggle { + flex: 0 0 auto; + background: none; + border: none; + cursor: pointer; + color: var(--text-faint); + font-size: 0.8rem; + line-height: 1; + padding: 4px 8px; + transition: transform 0.15s, color 0.15s; +} +.panel-toggle.open { transform: rotate(180deg); color: var(--text); } +.panel-toggle:hover { color: var(--accent); } + +.format-row.row-open, +.topic-list li.li-open .topic-row { background: var(--panel-soft); } + +.action-panel { + display: flex; + gap: 0.5rem; + padding: 0.15rem 0.75rem 0.55rem calc(0.75rem + 8px); +} + +.panel-btn { + flex: 1; + padding: 0.45rem 0.6rem; + border: 1px solid var(--border-strong); + border-radius: 6px; + background: var(--panel); + color: var(--text); + font-size: 0.82rem; + font-weight: 600; + cursor: pointer; + transition: all 0.15s; +} +.panel-btn:hover { border-color: var(--accent); } +.panel-btn:disabled { opacity: 0.4; cursor: not-allowed; } +.panel-btn:disabled:hover { border-color: var(--border-strong); } +.panel-btn.play { + color: var(--success); + background: var(--success-soft); + border-color: var(--success-border); +} +.panel-btn.play:hover { background: var(--success-soft-hover); } +.panel-btn.danger { color: var(--danger); } +.panel-btn.danger:hover { border-color: var(--danger); } +.panel-btn.armed { background: var(--danger); color: #fff; border-color: var(--danger); } + .format-x { display: none; color: var(--danger); @@ -915,20 +995,20 @@ function confirmDeleteProject(name) { .new-topic-toggle:hover { background: var(--accent-hover); } .new-topic-toggle.active { background: var(--accent-hover); } .thema-panel { - margin: 0.5rem 0; - padding: 0.75rem; + margin: 0.4rem 0; + padding: 0.5rem; background: var(--panel-soft); border: 1px solid var(--border); - border-radius: 10px; + border-radius: 8px; display: flex; flex-direction: column; - gap: 0.55rem; + gap: 0.35rem; font-size: 0.85rem; } .dlg-input, .dlg-textarea { width: 100%; box-sizing: border-box; - padding: 0.5rem 0.6rem; + padding: 0.4rem 0.5rem; border: 1px solid var(--border); border-radius: 6px; background: var(--panel); @@ -936,23 +1016,23 @@ function confirmDeleteProject(name) { font: inherit; } .dlg-input:focus, .dlg-textarea:focus { outline: none; border-color: var(--accent); } -.dlg-textarea { resize: vertical; min-height: 3rem; } -.dlg-sources { display: flex; gap: 0.4rem; } +.dlg-textarea { resize: vertical; min-height: 2rem; } +.dlg-sources { display: flex; gap: 0.3rem; } .dlg-sources button { flex: 1; - padding: 0.45rem 0.2rem; + padding: 0.35rem 0.2rem; border: 1px solid var(--border); border-radius: 6px; background: var(--panel-soft); color: var(--text); cursor: pointer; - font-size: 0.82rem; + font-size: 0.8rem; } .dlg-sources button.active { background: var(--accent); border-color: var(--accent); color: var(--on-accent); } .dlg-hint { margin: 0; font-size: 0.75rem; color: var(--text-faint); } -.dlg-actions { display: flex; justify-content: flex-end; gap: 0.5rem; margin-top: 0.3rem; } +.dlg-actions { display: flex; justify-content: flex-end; gap: 0.4rem; margin-top: 0.1rem; } .dlg-actions button { - padding: 0.45rem 0.9rem; + padding: 0.4rem 0.8rem; border: 1px solid var(--border); border-radius: 6px; background: var(--panel-soft); @@ -962,4 +1042,10 @@ function confirmDeleteProject(name) { .dlg-create { background: var(--accent); border-color: var(--accent); color: var(--on-accent); } .dlg-create:disabled { opacity: 0.45; cursor: default; } +/* Thema-Edit-Panel innerhalb der Themen-Liste */ +.edit-panel { margin: 0 0.75rem 0.5rem; } +.dlg-delete { margin-right: auto; color: var(--danger); } +.dlg-delete.armed { background: var(--danger); border-color: var(--danger); color: #fff; font-weight: 700; } +select.dlg-input { cursor: pointer; } + diff --git a/templates/Prompt/Relevanz-Mapping.md b/templates/Prompt/Relevanz-Mapping.md new file mode 100644 index 0000000..fe7c64e --- /dev/null +++ b/templates/Prompt/Relevanz-Mapping.md @@ -0,0 +1,18 @@ +Mehrere Agenten haben die Subbausteine des Themas "{topic}" nach Relevanz bewertet (relevant/rand). Bei einigen sind sie sich uneinig. Entscheide für jeden strittigen Subbaustein die finale Relevanz. + +STRITTIGE SUBBAUSTEINE (mit den abgegebenen Stimmen): +{strittig} + +Relevanz: +- **relevant**: Kern-/Standardwissen, ein ernsthafter Lerner braucht es. +- **rand**: Randnotiz, Spezial-/Nischenfall, „nice to know" — nur für den Komplett-Guide. + +Regeln: +- Wäge die Stimmen ab, entscheide fachlich. Im Zweifel **relevant**. +- Für JEDE strittige Nummer genau einen Wert. + +Schreibe NUR die JSON-Datei nach: {out_path} + +Format (kein weiterer Text): +{{"relevanz": {{"1": "relevant", "4": "rand"}}}} +{extra} diff --git a/templates/Prompt/Relevanz-Recherche.md b/templates/Prompt/Relevanz-Recherche.md new file mode 100644 index 0000000..f6998d3 --- /dev/null +++ b/templates/Prompt/Relevanz-Recherche.md @@ -0,0 +1,19 @@ +Bewerte jeden Subbaustein des Themas "{topic}" nach Relevanz: relevant oder rand. Die Relevanz steuert, ob ein Punkt in die fokussierten Guides kommt oder nur in den Komplett-Guide. + +SUBBAUSTEINE (Nummer in Klammern der zugehörige Baustein): +{subbausteine} + +Relevanz: +- **relevant**: Kern- oder Standardwissen des Themas. Ein ernsthafter Lerner braucht es im normalen Gebrauch. Was man wirklich kennen sollte. +- **rand**: Randnotiz, Spezial-/Nischenfall, „nice to know", Deprecated/Exotik. Für die meisten Lerner verzichtbar — gehört nur in einen Komplett-Guide. + +Regeln: +- Bewerte jeden Punkt einzeln, gemessen an SEINEM Baustein und am Thema. +- Im Zweifel **relevant** — lieber zu viel als versehentlich Wichtiges weglassen. +- Urteile nur — erfinde nichts dazu, ändere keine Subbausteine. + +Schreibe NUR die JSON-Datei nach: {out_path} + +Format (für JEDE Nummer genau ein Wert; kein weiterer Text): +{{"relevanz": {{"1": "relevant", "2": "rand", "3": "relevant"}}}} +{extra} diff --git a/uni/aak (Copy)/Skript.pdf b/uni/aak (Copy)/Skript.pdf new file mode 100644 index 0000000..030f074 --- /dev/null +++ b/uni/aak (Copy)/Skript.pdf @@ -0,0 +1,4456 @@ +%PDF-1.3 +% +1 0 obj +<< /Metadata 3 0 R /Pages 4 0 R /Type /Catalog >> +endobj +2 0 obj +<< /Author () /CreationDate (D:20250704113420+00'00') /Creator (LaTeX with hyperref) /ModDate (D:20250704113420+00'00') /Producer (LuaTeX-1.16.0) >> +endobj +3 0 obj +<< /Subtype /XML /Type /Metadata /Length 808 >> +stream + + + + + + + + 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[<8b41d259c856cf2ef2f3fd39b8622ab8><8b41d259c856cf2ef2f3fd39b8622ab8>] >> +startxref +714700 +%%EOF diff --git a/uni/aak (Copy)/Skript.txt b/uni/aak (Copy)/Skript.txt new file mode 100644 index 0000000..371697a --- /dev/null +++ b/uni/aak (Copy)/Skript.txt @@ -0,0 +1,2562 @@ +6 Einführung in die Komplexitätstheorie +In den vorherigen Kapiteln haben wir bereits einige theoretische Probleme kennenge- +lernt. Wir haben insbesondere gesehen, dass sich einige Probleme schneller lösen (bzw. +approximieren) lassen als andere. Außerdem scheint es Probleme zu geben, bei denen +man schnell den Mut verliert, eine exakte Lösung in Polynomialzeit zu berechnen. Diese +Probleme scheinen schwerer zu sein als andere. Was ist die Schwere eines Problems? Die +Komplexitätstheorie befasst sich noch weitergehend mit der Frage nach der Komplexität +von Problemen, dh. man versucht zu klassifizieren wie schwer Probleme sind. + In diesem Kapitel werden wir die Komplexitätsklassen P und N P einführen und wol- +len dann die Begriffe der N P -schwere und N P -vollständigkeit verstehen. Im Anschluss +werden wir einige Beispiele N P -vollständiger Probleme studieren. + + +6.1 Einleitung: Cliquenproblem +Als erstes Beispiel betrachten wir das sogenannte Cliquenproblem auf Graphen. Eine +Clique ist eine Teilmenge C ⊆ V , wobei {u, v} ∈ E für alle u, v ∈ C mit u 6= v. Das +Cliquenproblem können wir nun wie folgt angeben: + +Gegeben: Ein ungerichteter Graph G = (V, E) und eine Zahl k ≥ 1. + +Entscheide: Hat der Graph G eine Clique C ⊆ V mit mindestens k Knoten (d.h. mit + |C| ≥ k)? + +Dies ist ein sogenanntes Entscheidungsproblem, welches also als Antwort Ja oder Nein +gibt. + +Beispiel 6.1. Man betrachte den folgenden Graphen G = (V, E) mit V = {1, 2, 3, 4, 5} +und E = {{1, 2}, {1, 3}, {2, 3}, {3, 4}, {3, 5}}: + + 2 + b + + + + + 1 b b + 3 + + b b + 4 5 +Hier ist C = {1, 2, 3} eine Clique der Größe 3, aber C = {2, 3, 4} ist keine Clique. + + + +200 + 6.1 Einleitung: Cliquenproblem + + +Offensichtlich löst der folgende Algorithmus das Problem exakt: + +Algorithmus: + Teste für alle V 0 ⊆ V mit |V 0 | = k, ob V 0 eine Clique bildet. + +Wir bezeichnen mit T (G, k) seine Laufzeit für eine Eingabe (G, k). +Lemma 6.2. Der obige Algorithmus hat im schlimmsten Fall mindestens exponentielle +Laufzeit; d.h. T (G, k) ≥ 2|V |/2 für k = |V |/2. +Beweis. Für |V | = n, k = n/2 und n gerade: Der Test für jedes V 0 erfordert Ω(k 2 ) +Schritte. Es gibt nk viele k-elementige Teilmengen. Dann erhalten wir hier mit +   + n n(n − 1) · (n/2 + 1) + = ≥ 2n/2 + n/2 n/2(n/2 − 1) · 1 +mindestens eine exponentielle Laufzeit. + +Frage: Gibt es schnellere (bzw. insbesondere polynomielle) Algorithmen? +Definition 6.3. (i) Ein Alphabet Σ ist eine endliche Menge an Symbolen. Σ∗ ist die + Menge aller endlichen Wörter über Σ. + (ii) Für jedes x ∈ Σ∗ bezeichne |x| die Länge von x. +Bemerkung 6.4. Jeder Algorithmus arbeitet auf einem Alphabet (d.h. seine Eingaben +und Ausgaben sind Elemente aus Σ∗ ). Für einen Algorithmus A, bezeichnet A(x) die +Ausgabe zur Eingabe x ∈ Σ∗ . + Die Laufzeit eines Algorithmus A zur Eingabe x (gegeben durch die Anzahl der aus- +geführten Operationen) wird durch TA (x) bezeichnet. Die Worst Case Laufzeit zu Ein- +gaben der Größe n ist + TA (n) = max{TA (x)|x ∈ Σ∗ , |x| = n}. +Definition 6.5. (i) Eine Teilmenge L ⊆ Σ∗ heißt Sprache. + (ii) Ein Entscheidungsproblem ist ein Tripel (L, U, Σ) wobei L, U Sprachen sind und + L ⊆ U. +(iii) Ein Algorithmus A löst das Entscheidungsproblem (L, U, Σ) (oder auch A entschei- + det die Sprache L bezüglich U ), wenn für alle x ∈ U gilt: + – x ∈ L impliziert A(x) = 1, + – x 6∈ L impliziert A(x) = 0. + Häufig ist U = Σ∗ . Dann sagt man, dass A die Sprache L ⊆ Σ∗ entscheidet. Das +Cliquenproblem etwa können wir als Sprache (Menge von Wörtern) wie folgt darstellen: + CLIQU E = {u#v | u ∈ {0, 1}∗ stellt Adjazenzmatrix von G dar, + v ∈ {0, 1}∗ stellt k dar, + G hat Clique mit k Knoten }. + + + 201 + 6 Einführung in die Komplexitätstheorie + + +Hierbei ist Σ∗ = {0, 1}∗ die Menge aller endlichen Wörter (inklusive dem leeren Wort ) +über dem Alphabet Σ = {0, 1}. + +Bemerkung 6.6. u ist ein Wort der Länge O(|V |2 ), v ist ein Wort der Länge O(log |V |). + + +6.2 Komplexitätsklassen P, N P +Definition 6.7. Die Menge aller polynomiell entscheidbaren Sprachen bzw. Entschei- +dungsproblemen wird mit P bezeichnet: + + P = {L ⊆ Σ∗ | es existiert ein Algorithmus A, der L in Laufzeit + TA (n) = O(nd ) mit d = O(1) entscheidet}. + + Neben P gibt es unter anderem auch die Klasse N P . Diese wird häufig über nicht- +deterministische Algorithmen eingeführt (bei denen es zu jeder Eingabe eine endliche +Menge von Rechenwegen gibt). + +Definition 6.8. Ein nicht-deterministischer Algorithmus entscheidet eine Sprache L, + + • wenn es für jedes Wort w ∈ L mindestens einen Rechenweg gibt, der w akzeptiert, + + • wenn jeder akzeptierende Rechenweg in polynomieller Zeit läuft und + + • wenn für jedes Wort w 6∈ L jeder Rechenweg w ablehnt. + + Die Anzahl der Rechenwege kann dabei exponentiell von der Eingabe abhängen. + + Die Klasse N P ist nun genau die Menge aller Sprachen, die durch einen nicht-deterministischen +Algorithmus in Polynomialzeit entschieden werden können. Wir bedienen uns hier aller- +dings einer alternativen Definition der Klasse N P über polynomielle Verifizier. Die so +definierte Klassen von Problemen sind aber gleich. + +Definition 6.9. Es sei L ⊆ Σ∗ . Ein (deterministischer) Algorithmus, der auf Σ∗ × Σ∗ +arbeitet, heißt ein Verifizierer für L, wenn + + L = {x ∈ Σ∗ |∃c ∈ Σ∗ mit A(x, c) = 1}. + + Hierbei nennt man ein Wort c mit A(x, c) = 1 ein Zertifikat für x (bzw. für die +Aussage x ∈ L). + +Definition 6.10. Ein Algorithmus A ist ein polynomieller Verifizierer für eine Sprache +L, wenn eine Konstante d existiert, so dass es für jedes x ∈ L ein Zertifikat c gibt mit +TA (x, c) = O(|x|d ). + +Definition 6.11. Die Klasse aller polynomiell verifizierbaren Sprachen wird mit N P +bezeichnet; d.h. + + N P = {L ⊆ Σ∗ | es existiert ein polynomieller Verifizierer für L}. + + + +202 + 6.2 Komplexitätsklassen P, N P + + +Satz 6.12. Es gilt P ⊆ N P . + +Beweis. Zu einem Entscheidungsproblem L aus P gibt es einen Algorithmus A, der +L entscheidet. Diesen Algorithmus kann man als Verifizierer auffassen, der die zweite +Eingabe c einfach ignoriert. Damit ist L auch in N P . + + Nun können wir auf das Cliquenproblem zurückkommen und die folgende Aussage +beweisen. + +Lemma 6.13. Das Entscheidungsproblem k-Clique ist in N P . + +Beweis. Durch Angabe eines polynomiellen Verifizieres für k-Clique. Der Algorithmus +bekommt neben der Eingabe (einen Graph G = (V, E) und eine Zahl k) ein Zertifikat +(u.z. eine Teilmenge C ⊆ V ). Der Algorithmus testet dann, ob C eine Clique der Größe +≥ k ist. Der Test geht wie folgt: + + (1) prüfe für alle zweielementigen Teilmengen {u, v} ⊆ C, ob {u, v} ∈ E liegt. + + (2) gebe Ausgabe Ja aus, wenn jeder Test in (1) positiv ausfiel und |C| ≥ k ist. + Ansonsten Ausgabe Nein. + + Schritt (1) geht in Zeit O(|C|2 ) = O(|V |2 ) und die Kardinalitätsprüfung in (2) in +O(|C|) = O(|V |) Zeit. Da die Eingabelänge Ω(|V | + |E|) ist, ist die Laufzeit des Algo- +rithmus polynomiell. + + Nachdem wir nun gesehen haben, wie die Klassen P und N P definiert haben, können +wir den Begriff der Schwere genauer beleuchten. Dafür ist es notwendig zu verstehen, +wann ein Problem auf ein anderes Problem reduziert werden kann. Wir benötigen die +folgende Definition. + +Definition 6.14. Es seien L1 ⊆ Σ∗1 , L2 ⊆ Σ∗2 zwei Sprachen. + + (i) Eine Funktion f : Σ∗1 → Σ∗2 mit der Eigenschaft + + w ∈ L1 ⇔ f (w) ∈ L2 ∀w ∈ Σ∗1 + + heißt Transformation von L1 auf L2 . + + (ii) Eine solche Transformation heißt polynomielle Transformation, wenn es ein Po- + lynom p und einen Algorithmus A gibt, der für alle w ∈ Σ∗1 die Ausgabe f (w) der + Länge ≤ p(|w|) in (polynomieller) Zeit TA (w) ≤ p(|w|) berechnet. + +Dann heißt L1 polynomiell reduzierbar auf L2 . Als Notation verwenden wir L1 ≤ L2 . + +Endlich können wir den Begriff der N P -Schwere bzw. N P -Vollständigkeit formal fassen: + +Definition 6.15. (i) Eine Sprache L0 heißt N P -schwer, g.d.w. ∀L ∈ N P : L ≤ L0 . + + (ii) Eine Sprache L0 heißt N P -vollständig, g.d.w. L0 ∈ N P und ∀L ∈ N P gilt: L ≤ L0 . + + + + 203 + 6 Einführung in die Komplexitätstheorie + + +Eine fundamentale Eigenschaft der N P -vollständigkeit ist die folgende: + +Satz 6.16. Es sei L0 N P -vollständig. Dann gilt P = N P , g.d.w. L0 ∈ P . + +Beweis. ⇒: Es sei P = N P . Zeige dann, dass L0 ∈ P . + L0 ist NP-vollständig. Dann gilt insbesondere L0 ∈ N P . Wegen P = N P gilt dann +L0 ∈ P . + ⇐: Es sei L0 ∈ P . Zeige hier, dass P = N P . + Da L0 ∈ P ist, gibt es einen Algorithmus A0 , der L0 in Polynomzeit T0 entscheidet. +Es sei nun L ∈ N P beliebig gewählt. Zeige: L ∈ P . + Da L0 N P -vollständig ist und L ∈ N P , gilt L ≤ L0 . Sei also Af ein Algorithmus, +der die polynomielle Transformation f von L auf L0 in Polynomzeit Tf berechnet mit +u ∈ L ⇔ f (u) ∈ L0 . + Wir konstruieren A aus A0 und Af durch Hintereinanderschalten; siehe folgendes Bild: + + + Af A0 ja + u f (u) + nein + + Zeit Tf (|u|) Zeit T0 (|f (u)|) + + Dann akzeptiert A ein Wort u, g.d.w. A0 das Wort f (u) akzeptiert. Dies ist äquivalent +zu f (u) ∈ L0 bzw. u ∈ L. D.h. Algorithmus A entscheidet L mit Gesamtlaufzeit ≤ +Tf (|u|)+T0 (|f (u)|) ≤ T (|u|) für ein Polynom T . Hierbei verwende, dass |f (u)| ≤ poly(|u|) +und dass die Komposition von Polynomen wieder ein Polynom ergibt. + +Konsequenz von Satz 6.16: Wenn ein N P -vollständiges Entscheidungsproblem in P +liegt, dann sind alle N P Entscheidungsprobleme in P . + + Die gleiche Aussage gilt auch für ein N P -schweres Problem. +Offenes Problem der Komplexitätstheorie: + Gilt P = N P oder P 6= N P ? + + Dies ist eines der wichtigsten offenen Probleme der Informatik. Es wurde vom Clay +Mathematics Institute in die Liste der Millennium-Probleme aufgenommen; mit einem +Preisgeld von 1-Million Dollar: www.claymath.org. + +Bemerkung 6.17. Wenn L1 ≤ L2 und L2 ≤ L3 , dann gilt auch L1 ≤ L3 . + +Beweis. siehe Übung. + +Korollar 6.18. Ist L0 N P -vollständig, L0 ≤ L1 und L1 ∈ N P , so ist auch L1 N P - +vollständig. + + + +204 + 6.3 Definition von P und N P per Turingmaschine + + +Beweis. Zeige die folgende Behauptung: ∀L ∈ N P gilt L ≤ L1 . +Sei dazu L ∈ N P beliebig gewählt. Zeige dann die Aussage L ≤ L1 . Wegen der N P - +vollständigkeit von L0 und da L ∈ N P ist, gilt L ≤ L0 . Nach Voraussetzung gilt +zusätzlich L0 ≤ L1 . Da die Relation ≤ transitiv ist (siehe Bemerkung 6.17), gilt dann: +L ≤ L1 . + Da L1 ∈ N P wegen der Voraussetzung, ist daher L1 auch N P -vollständig. + + Das obige Ergebnis in Korollar 6.18 ermöglicht uns, N P -vollständige Probleme aus +bekannten N P -vollständigen Problemen zu konstruieren (bzw. deren N P -vollständigkeit +zu beweisen). Für diese Entscheidungsprobleme gibt es keine polynomiellen Algorithmen, +außer P = N P . + + +6.3 Definition von P und N P per Turingmaschine +Eine nicht-deterministische Turingmaschine (NDTM) hat die Form: + + M = (Q, Γ, q0 , δ, F ). + +Hierbei beschreibt Q die nicht-leere endliche Zustandsmenge, Γ das Arbeitsalphabet, +wobei Γ ⊃ Σ gilt (dabei ist Σ das Eingabealphabet). Die Zustandsmenge enthält den +Anfangszustand q0 ∈ Q und die Endzustandsmenge F ⊂ Q. Zuletzt beschreibt δ die +Übergangsfunktion: + δ : Q × Γ −→ 2Q×Γ×{−1,0,1} . +Es gilt, dass |δ(q, a)| = 0, 1, 2, . . . sein kann, man also mehrere mögliche Folgezustände +haben kann. + + Arbeitsfeld + + u a u + + + + + q + + + + Ein Übergang δ(q, a) → (q 0 , a0 , −1/1/0) besagt: im Zustand q mit a auf dem Arbeits- +feld (AF) drucke a0 auf AF, bewege den Schreiblesekopf (bzw. das AF) nach links, rechts +oder bleibe stehen und gehe danach in Zustand q 0 über. + Die möglichen Berechnungen einer NDTM wird über Konfigurationen (die die Band- +inschrift plus den Zustand enthalten) beschrieben, wobei eine Konfiguration c wie folgt +definiert ist: + + c ∈ Γ∗ × Q × Γ × Γ∗ + u q a v, + + + + 205 + 6 Einführung in die Komplexitätstheorie + + +wobei u und v Wörter sind, q ein Zustand und a der Buchstabe auf dem AF ist. + Für u = u0 b, v sei die Folgekonfiguration von uqav definiert durch: + + u0 q 0 ba0 v falls δ(q, a) = (a0 , 1, q 0 ) + u0 ba0 q 0 v falls δ(q, a) = (a0 , −1, q 0 ) + u0 bq 0 a0 v falls δ(q, a) = (a0 , 0, q 0 ) + +Als Notation verwenden wir C ` C 0 ; dabei ist C 0 die Folgekonfiguration von C. + +Berechnung zu Wörtern bzw. Eingaben: Hierbei ist die Startkonfiguration durch q0 [w +gegeben, wobei [ ein Sonderzeichen 'Blank' (bzw. ein Leerzeichen) darstellt mit [ ∈ Γ\Σ. +Eine Konfiguration C nennt man Stopkonfiguration, falls keine Folgekonfiguration von +C existiert. Eine Berechnung zu einer Eingabe w ∈ Σ∗ kann dann durch eine Folge +C0 , . . . , Ck von Konfigurationen dargestellt werden, wobei C0 die Startkonfiguration zu +einer Eingabe w ist, Ci ` Ci+1 , für alle i < k gilt und Ck eine Stopkonfiguration ist. +Eine Berechnung heißt genau dann akzeptierend, wenn der Zustand von Ck in F liegt. + Sei L ⊆ Σ∗ und T : N → N. M akzeptiert L [in nicht deterministischer Zeit T ] genau +dann wenn ∀w ∈ Σ∗ : (w ∈ L ⇐⇒ ∃ akzeptierende Berechnung von M zu w [mit Länge +≤ T (|w|)]. + Damit können wir die Zugehörigkeit zu der Komplexitätsklasse NP nun definieren. + +Definition 6.19. Eine Sprache L ⊂ Σ∗ ist in NP ⇐⇒ ∃ eine NDTM M über Alphabet +Γ ⊃ Σ und ein Polynom T , so dass gilt: M akzeptiert L in nicht deterministischer Zeit +T. + +Deterministische Turingmaschine (DTM) Falls |δ(q, a)| ≤ 1 ∀(q, a) ∈ Q × Γ, es also +höchstens einen gültigen Übergang aus jedem Zustand gibt, nennt man eine Turingma- +schine deterministisch. Die Berechnung der DTM ist analog zu NDTMs über Konfigu- +rationen definiert. + Eine DTM M entscheidet L [in deterministischer Zeit T ] genau dann wenn für alle +Wörter w ∈ Σ∗ die Berechnung von w mit einer Stopkonfiguration endet [und Länge +≤ T (|w|) hat]. Dabei liegt der Zustand der Stopkonfigurationen in F , genau dann wenn +w in L liegt. + Die Zugehörigkeit zu der Komplexitätsklasse P wird nun analog definiert. + +Definition 6.20. L ⊂ Σ∗ ∈ P ⇐⇒ ∃ eine DTM M über Alphabet Γ ⊃ Σ und ein Poly- +nom T , so dass M die Sprache L in Zeit T entscheidet. + + Nun bleibt zu zeigen, dass beide gezeigten Definitionen der Komplexitätsklasse N P +äquivalent sind, sie also die gleiche Klasse beschreiben. + +Satz 6.21. Die Klassen der Sprachen bezüglich Verifizierer und NDTM sind äquivalent +und beschreiben beide die gleiche Klasse N P . + +Beweis. ” ⊃ ” : Sei N eine NDTM, die eine Sprache L in Zeit nc für eine Konstante +c entscheidet. Für jedes x ∈ L gibt es eine Folge nicht-deterministischer Übergänge, + + + +206 + 6.4 Satisfiability Problem (SAT) + + +so dass N bei Eingabe x einen akzeptierenden Zustand qacc erreicht. Diese Folge kann +durch einen String u der Länge nc über einem Alphabet σ 0 mit O(1) vielen Buchstaben +repräsentiert werden (hierbei hängt die Kardinalität von σ 0 von der maximalen Anzahl +von Verzweigungen (d.h. |δ(q, a)|) ab. Dann folgt direkt, dass es eine DTM M gibt, die bei +Eingabe < x, u > genau dieselben Berechnungen durchführt, also die gleichen Übergänge +wie N macht. Für die Eingabe x ∈ L folgt also auch, dass M einen akzeptierenden +Zustand erreichen muss. Dann ist M also auch ein Verifizierer für L. + ” ⊂ ” : Sei L ⊆ Σ∗ gegeben und sei M ein Verifizierer für L. Sei p : N → N ein +Polynom, das die Länge der Zertifikate angibt. Sei nc die Laufzeit der DTM M , wobei +c wieder eine Konstante ist. Eine NDTM N , die w in Polynomzeit entscheidet, kann bei +Eingabe x ∈ Σ∗ wie folgt vorgehen: + Phase (1): Nutze nicht-deterministische Entscheidungen um einen beliebigen String u +der Länge poly(|x|), also polynomiell in |x|, auf ein Arbeitsband zu schreiben. + Phase (2): Simuliere M bei Eingabe < x, u >. Dann folgt direkt, dass N in Polynomzeit +entscheidet, da M diese Eingabe in Polynomzeit verifizieren kann. + + +6.4 Satisfiability Problem (SAT) +Der Kanadier Stephen A. Cook begründete 1971 die Klasse der N P -vollständigen Proble- +me in N P , indem er zeigte, dass ein N P -vollständiges Problem existiert. Dieses Problem +ist das sogenannte Satisfiability Problem (SAT). Leonid Levin hatte einen vergleichbaren +Satz 1973 veröffentlicht. + +Satisfiability Problem (SAT) + + Das Satisfiability Problem (SAT) formalisiert die Frage nach der Erfüllbarkeit eines +aussagenlogischen Ausdrucks. Aussagenlogische Ausdrücke sind zusammengesetzt aus +Variablen und den elementaren Operatoren/Junktoren ∧ (und), ∨ (oder) und ¬ (nicht). +Die Variablen sind Boolesche Variablen und wir notieren sie als xi, dh. x0, x1, x10, x11, . . . +und so weiter. Für jede Variable xi existieren zwei Literale. Diese sind das positive Lite- +ral xi und das negative Literal ¬xi. Aus den Literalen setzt man mit dem elementaren +Operator ∨ (oder) sogenannte Klauseln zusammen, dh. disjunktive Ausdrücke der Form +(y1 ∨ · · · ∨ yk ) mit Literalen y1 , . . . , yk . + Ein Ausdruck α = C1 ∧ . . . ∧ Cm mit Klauseln C1 , . . . , Cm heißt Ausdruck in Kon- +junktiver Normalform (KNF). Ein Ausdruck α heißt erfüllbar, g.d.w. eine Belegung φ +der Variablen in α mit true, f alse existiert, so dass α unter der Belegung φ zu true +ausgewertet wird (siehe Beispiel 6.23). + Nun können wir das SAT-Problem als Sprache wie folgt angeben: + + SAT = {α | α Boolescher Ausdruck in KNF, α erfüllbar } +Bemerkung 6.22. SAT ⊂ {x, 1, 0, ∨, ∧, ¬, (, )}∗ . +Beispiel 6.23. Betrachte α = (x1∨x10)∧(x1∨¬x10)∧(x10). Wir erhalten die folgende +Wahrheitstabelle: + + + + 207 + 6 Einführung in die Komplexitätstheorie + + + x1 x10 α + false false false + true false false + false true false + true true true + +Die Belegung ψ(x1) = ψ(x10) = true ist eine erfüllende Belegung. + +Satz 6.24. (Cook, 1971/Levin, 1973): SAT ist N P -vollständig. + +Beweis. SAT ∈ NP Wir skizzieren dies für eine entsprechende nichtdeterministische Tu- + ringmaschine mit den Zwischenergebnissen auf dem Turingband in Abb. 6.1. Salopp + gesagt raten“ wir nichtdeterministisch eine Belegung der Variablen und prüfen + ” + dann, ob sie den Ausdruck erfüllt. + + α + (1) ↓ det. + α#Liste der auftretenden Variablen + (2) ↓ nichtdet. + α#Belegung der Variablen durch wahr/falsch: + x101 → T 101 oder F 101. + (3) ↓ det. + α#Wert von α + (4) ↓ det. + F , wenn α = wahr; dummy-Zustand, falls α = falsch. + + Abb. 6.1: SAT ∈ NP: Alle vier Schritte sind in polynomieller Zeit ausführbar. + + +L ∈ NP =⇒ L ≤ SAT Sei M eine NDTM und akzeptiere L in nichtdeterministischer + Zeit T (für ein geeignetes Polynom T ). Wir suchen eine Transformation u → αu , + die polynomzeitberechenbar ist mit u ∈ L ⇐⇒ αu ∈ SAT. + Idee: αu wird so beschrieben, dass M für u in genau T (u) Schritten eine akzeptie- + rende Stoppkonfiguration erreicht. Sei also + + M = (Q, Γ, q0 , δ, F ), + + wobei Q = {q0 , . . . , qs }, F = {qr , . . . , qs }, Γ = {a0 , . . . , am }, a0 = [. Ohne Ein- + schränkung der Allgemeinheit gelte: Akzeptiert M das Wort u = aj1 . . . aj|u| mit + einer Berechnung der Länge höchstens T (|u|), so akzeptiere M das Wort mit einer + Berechung der Länge genau gleich T (|u|). + Wir stellen δ durch eine Folge von Zeilen (Fünftupeln) z1 , . . . , z% der Form qaq 0 a0 β + dar (für (q 0 , a0 , β) ∈ δ(q, a)). O.B.d.A betrachten wir nur Turingmaschinen, die + nicht links vom Feld 0 arbeiten. Der zu konstruierende Ausdruck in KNF α enthält + Variablen, deren Bedeutung in Tabelle 6.1 zusammengefasst ist. + + + +208 + 6.4 Satisfiability Problem (SAT) + + + Variablen in αk Bedeutung (wahr, falls. . . ) + ztk nach t Schritten wird Zustand qk erreicht + sti nach t Schritten ist Feld Nr. i das Arbeitsfeld + btl nach t Schritten wird Zeile zl ausgeführt + atij nach t Schritten steht auf Feld Nr. i der Buchstabe aj , +Hierbei ist 0 ≤ t, i ≤ T (|u|), 0 ≤ k ≤ S, 1 ≤ l ≤ %, 0 ≤ j ≤ m. + Tab. 6.1: Variablen in αk + + + + Die Anzahl der Variablen ist nach oben durch c · T (|u|)2 beschränkt, wobei c eine + von der Turingmaschine abhängige Konstante ist. + Wir bauen nun einen Ausdruck αu auf mit + + αu = αAnfang ∧ αEnde ∧ αEindeutig ∧ αÜbergang . + + Das Ziel ist, dass erfüllende Belegungen von αu den akzeptierenden Berechnungen + in Zeit T (|u|) von M für u = aj1 . . . aj|u| entsprechen sollen. Wir setzen + + αAnfang = z00 ∧ s00 ∧ a000 ∧ a01j1 ∧ a02j2 ∧ · · · ∧ a0|u|j|u| ∧ a0(|u|+1)0 ∧ · · · ∧ a0 T (|u|) 0 + αEnde = zT (|u|) r ∨ · · · ∨ zT (|u|) s + genau ein Zustand“ + ^ + αEindeutig = + ” + 0≤t≤T (|u|) + + ∧ genau ein Arbeitsfeld“ + ” + ∧ für alle Felder i = 0, . . . , T (|u|) genau ein Buchstabe“ +  + ” + Den ersten Teil, genau ein Zustand“, schreiben wir als + ” + ^ + (zt0 ∨ · · · ∨ zts ) ∧ ¬(zti ∧ ztj ), + | {z } + i6=j + (¬zti ∨¬ztj ) + + die anderen Bedingungen analog. Wir setzen + ^ + αÜbergang = αt , + 0≤t≤T (|u|) + + wobei αt Folgendes modelliert: + • Buchstaben auf nicht-Arbeitsfeldern bleiben unverändert + • Falls das Arbeitsfeld Nr. i hat und die l. Zeile qkl ajl qk̃l aj̃l βl ausgeführt wird, + gilt: + a) nach t Schritten wird qkl erreicht + b) nach t Schritten steht auf Feld i der Buchstabe aj l + c) nach t + 1 Schritten wird qk̃l erreicht + + + + 209 + 6 Einführung in die Komplexitätstheorie + + + d) nach t + 1 Schritten steht auf Feld i Buchstabe aj̃l + e) nach t + 1 Schritten hat das Arbeitsfeld die Nummer i + βl . + Wir erhalten insgesamt: + ^ ^  + αt = ((¬sti ) ∧ atij ) → a(t+1)ij + 0≤i≤T (|u|) 0≤j≤m + ^  + ∧ (sti ∧ btl ) → (ztkl ∧ atijl ∧ z(t+1)k̃l ∧ a(t+1)i̃l ∧ s(t+1)(i+βl ) ) + 1 0 : ∃x0 ≥ 0 : ∀x ≥ xo : |f (x)| ≤ c · |g(x)| + +Man kann dies auch als fg(x) + (x) + → 0 für x → ∞. Nach dieser Definition gilt also auch +dass δn = o(n), weswegen die beiden Schreibweisen austauschbar verwendet werden. Die +Originale Formulierung der ETH verwendet die δn Notation, weswegen sie in unserer +Formulierung der Theoreme verwendet wurde. + + +6.14 Hamiltonkreis Problem +Das Hamitonkreis Problem (HK) is wie folgt definiert: + +Gegeben: Ein ungerichteter Graph G = (V, E) mit V = {v1 , . . . , vn }. +Entscheide: Gibt es einen Hamiltonschen Kreis; d.h. eine Permutation π der Knoten + (vπ(1) , . . . , vπ(n) ) mit {vπ(i) , vπ(i mod n+1) } ∈ E für alle i = 1, . . . , n? + +Beispiel 6.41. Der Graph in dem folgenden Bild hat einen entsprechenden Hamilton- +kreis; dieser ist fettgezeichnet. + + + +220 + 6.14 Hamiltonkreis Problem + + + + +Satz 6.42. Das Hamiltonkreis Problem ist NP-vollständig. + +Bemerkung 6.43. Die gleiche Aussage gilt auch für gerichtete Graphen. + +Beweis. (a) HK is in NP: Wähle als Zertifikat eine Permutation π der Knoten und teste, +ob {vpi(i) , vπ(i+1 mod n) } ∈ E ist für alle i = 1, . . . , n. + (b) 3−SAT 0 ≤ HK: Gegeben seine SAT Formel mit genau 3 Literalen pro Klausel (ist +ebenfalls NP-vollständig). Zur Beweisidee geben wir die folgende Graph-Konstruktionen +an. Als erstes verwende die folgende A-Komponente: + + 1 3 + + + + + 5 6 7 8 + + + + + 2 4 + + + + + Dann gibt es zwei Möglichkeiten, die Komponente A zu durchlaufen; siehe die folgen- +den zwei Abbildungen: + + + 1 3 + + + + + 2 4 + + + + + 1 3 + + + + + 2 4 + + + + + Für jede Klausel verwenden wir die folgende Komponente B: + + + + 221 + 6 Einführung in die Komplexitätstheorie + + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + Eine der vielen Möglichkeiten, wie B durchlaufen werden kann, ist im folgenden Bild +dargestellt. + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + Insgesamt gibt es für jede nicht-leere Teilmenge der rechten Kanten {u1 , u2 }, {u2 , u3 }, {u3 , u4 } +eine Möglichkeit die Komponente B zu durchlaufen. Bei einer leeren Teilmenge dieser +Kanten funktioniert dies aber nicht; dies entsprecht dem Fall, dass alle Literale falsch +gewählt worden sind. Als Kurzform für die Spezialkomponenten A und B verwenden +wir: + + 1 3 + + + 5 + + + 2 4 + + + + +222 + 6.14 Hamiltonkreis Problem + + 1 + + + + + 2 + + + 5 + + + 3 + + + + + 4 + + + + Im folgenden Beispiel zeigen wir die Konstruktion des Graphen G zu einem Booleschen +Ausdruck F . + +Beispiel 6.44. Ausdruck F = (x1 ∨ ¬x2 ∨ x3 ) ∧ (¬x1 ∨ x2 ∨ ¬x3 ) ∧ (¬x1 ∨ ¬x2 ∨ x3 ) und +der zugehörige konstruierte Graph G: + + + 13 + 1 + 10 + 2 + 14 + 3 4 + 15 + + 11 5 + 6 16 + + 7 17 + + 8 + 18 + 12 9 + + + + + Für die Variablensetzung von Variable xi wählt man im rechten Pfad im obigen Bild +jeweils zwischen zwei Knoten vi , wi für i = 1, . . . , n die rechte oder linke Kopie der +Kante {vi , wi }. Die A-Komponenten werden nun genutzt um die Verbindung zwischen +den Variablen und den Literalen in den B-Komponenten zu den entsprechenden Klauseln +herzustellen. Dazu bezeichnen wir die u Knoten in der B-Komponente zu Klausel Fi mit +ui1 , ui2 , ui3 und ui4 . + Wenn das j.te Literal in Klausel Fi gleich xk ist, dann verbinden wir die Kante +{uij , uij+1 } via einer A-Komponente mit der linken Kopie von {vk , wk }. Wenn das j.te +Literal gleich ¬xk , so verbinden wir die Kante {uij , uij+1 } mit der rechten Kopie. + Formal beweist man Ende die folgende Aussage: F is erfüllbar genau dann, wenn G +einen Hamiltonkreis besitzt. + + Die Details findet man im Buch von Papadimitriou und Steiglitz. + + + + 223 + 6 Einführung in die Komplexitätstheorie + + +Übungsaufgaben +Übung 6.45. Die Eingabe des Problems SAT3 ist ein boolscher Ausdruck in KNF, für +den entschieden werden soll, ob es mindestens 3 erfüllende Belegungen für diese Formel +gibt. + Zeigen Sie, dass SAT3 N P -vollständig ist. +Übung 6.46. Eine NAE-k-SAT (Not-All-Equal) Formel hat folgende Form m i i + V + i=1 (z1 , . . . , zk ), +wobei eine Klausel (z1i , . . . , zki ) mit Literalen z1i , . . . , zki genau dann erfüllt ist, wenn min- +destens ein Literal zu wahr und mindestens ein Literal zu falsch ausgewertet wird. + Beispiel: Für die NAE-3-SAT Formel (x, y, z) ∧ (¬x, y, z) ist durch x = y = z = true +keine erfüllende Belegung gegeben, da in diesem Fall die erste Klausel nicht erfüllt ist. +Bei der Belegung x = y = true und z = false hingegen ist die Formel erfüllt. + Zeigen Sie in einem ersten Schritt, dass NAE-4-SAT NP-vollständig ist, und in einem +zweiten Reduktionsschritt zeigen, dass ebenfalls NAE-3-SAT NP-vollständig ist. +Übung 6.47. Ein Independent Set in einem Graphen G = (V, E) ist eine Menge von +Knoten I, so dass für je zwei Knoten i, j ∈ I gilt, dass {i, j} 6∈ E. Für das Problem +Independent Set ist ein Graph G = (V, E) sowie eine Zahl k ∈ N gegeben und es soll +entschieden werden, ob es in G ein Independent Set mit k Knoten gibt. + Zeigen Sie, dass das Problem Independent Set N P -vollständig ist. +Übung 6.48. Ein Vertex Cover in einem Graphen G = (V, E) ist eine Menge von +Knoten C ⊆ V so dass für jede Kante {v, w} ∈ E gilt das v ∈ C oder w ∈ C. Für das +Problem Vertex Cover ist ein Graph G = (V, E) sowie eine Zahl k ∈ N gegeben und +es soll entschieden werden, ob es in G ein Vertex Cover mit k Knoten gibt. + Zeigen Sie, dass das Problem Vertex Cover N P -vollständig ist. +Übung 6.49. In dem Problem Clique-Member ist ein Graph G = (V, E), ein Knoten +v ∈ V sowie eine Zahl k ∈ N gegeben und es soll entschieden werden, ob es eine Clique +mit k Knoten gibt, die den Knoten v enthält. + Zeigen Sie, dass Clique-Member N P -vollständig ist. +Übung 6.50. In dem Problem Clique-Nomember ist ein Graph G = (V, E), ein +Knoten v ∈ V sowie eine Zahl k ∈ N gegeben und es soll entschieden werden, ob es eine +Clique mit k Knoten gibt, die den Knoten v nicht enthält. + Zeigen Sie, dass Clique-Nomember N P -vollständig ist, indem Sie eine Reduktion +von Clique auf Clique-Nomember angeben. +Übung 6.51. In dem Problem Z-Clique ist ein zusammenhängender Graph G = +(V, E) und eine Zahl k ∈ N gegeben und es soll entschieden werden, ob G eine Clique +mit k Knoten enthält. + Zeigen Sie, dass Z-Clique N P -vollständig ist. +Übung 6.52. Zeigen Sie, dass das Partition Problem ebenfalls NP-schwer ist, wenn +eine Teilmenge A00 ⊆ A gesucht ist mit + X X + a=3 a. + a∈A00 a∈A\A00 + + + + +224 + 6.14 Hamiltonkreis Problem + + +Übung 6.53. Für das Problem Subset Sum Cardinality seien n ganze Zahlen +c1 , . . . , cn ∈ N>0 (wobei n gerade sei) sowie eine Zahl K ∈ N gegeben. Es + P soll entschieden +werden, ob es eine Teilmenge S ⊂ {1, . . . , n} gibt mit |S| = n/2 und i∈S ci = K. + Zeigen Sie, dass das Problem Subset Sum Cardinality N P -vollständig ist. + +Übung 6.54. Eine Eingabe des Problems Even-Knapsack ist eine Menge von Items +I, sowie ein Rucksack mit gerader Größe B (d.h es gibt b ∈ N mit B = 2b), sowie +ein Profitwert P ∈ N. Jedes Item hat eine Größe si ∈ N und einen P Profit pi ∈ N. +Das Entscheidungsproblem ist eine Teilmenge I 0 ⊆ I zu finden mit + i∈I 0 si ≤ B und + i∈I 0 pi ≥ P . +P + Zeigen Sie, dass Even-Knapsack N P -vollständig ist. + +Übung 6.55. Die Probleme Multiple-Choice-Knapsack (kurz MC-Knapsack) ist wie folgt +definiert: Gegeben sind endliche Mengen C1 , . . . Ck ⊆ N × N von Items, eine Kapazität +B ∈ N und ein Zielprofit P ∈ N. Wir bezeichnen Ci als Klasse von Items und für ein +Item (w, p) ∈ Ci ist w das Gewicht und p der Profit des Items. Es soll entschieden +werden, ob aus jeder Klasse Ci genau ein Item (wi , pi ) gewählt werden kann, sodass die +gewählten Items gemeinsam + P den Zielprofit erfüllen ohne die Kapazität zu überschreiten, +d.h. i∈[k] pi ≥ P und i∈[k] wi ≤ B. + P + Das Problem ist in N P . Zeigen Sie: MC-Knapsack ist N P -vollständig + + + + + 225 + 7 Approximative Algorithmen +Das letzte Kapitel hat uns Einblicke in die Schwierigkeit der polynomiellen Berechen- +barkeit gegeben. Wir wissen nun insbesondere, dass es viele Probleme gibt, deren exakte +Lösung eine große, nicht-polynomielle Laufzeit benötigt - es sei denn, es gilt P = N P . +In Ermangelung einer Eingebung zu der Frage, ob P = N P oder P 6= N P gilt, hat sich +die Disziplin der Approximativen Algorithmen entwickelt; anstatt eine exakte Lösung zu +verlangen, lässt man die Berechnung geringfügig schlechterer Lösungen zu, welche sich +aber deutlich schneller berechnen lassen. Die exakte Lösung wird in diesem Sinne nur +noch approximiert. + In diesem Kapitel wollen wir einige solche Approximationsalgorithmen studieren. + + +7.1 Traveling Salesman Problem +Gegeben: Eine Menge V = {1, . . . , n}, Distanzen d(i, j) ∈ Z+ ∪ {∞} für alle i, j ∈ + {1, . . . , n} + +Gesucht: Eine minimale Rundreise, die jeden Knoten genau einmal besucht. + +Formal: Eine Lösung P + ist gegeben durch eine Permutation Π von {1, . . . , n}, wobei die + Länge der Tour ni=1 d(Π(i), Π(i + 1)) mit Π(n + 1) = Π(1) ist. + +Beispiel 7.1. Betrachte den folgenden Graphen mit 4 Knoten: + + 1 b 1 b 2 + 2 + 1 1 + 2 + b b + + 4 1 3 + Eine optimale Rundreise ist [1, 2, 3, 4, 1] mit Länge 4. + +Satz 7.2. Das Entscheidungsproblem zum Traveling Salesman Problem, ob eine Rund- +reise mit Länge ≤ L existiert, ist NP-vollständig. + +Beweis. Dies folgt durch eine Reduktion vom Hamiltonkreis Problem: Wähle für jede +Kante {u, v} ∈ E den Wert d(u, v) = 1 und für alle anderen Paare {u, v} 6∈ E den Wert +d(u, v) = |V | + 1 sowie L = |V |. + + + +226 + 7.1 Traveling Salesman Problem + + + Durch Modifikation der Distanzen d(u, v) = ∞ für alle {u, v} 6∈ E, kann man sogar +zeigen, dass es keinen approximativen Algorithmus für das allgemeine TSP Problem mit +beschränkter Approximationsgüte gibt. + Deswegen betrachten wir den symmetrischer Fall des TSP mit d(i, j) = d(j, i) und +mit Erfüllung der Dreiecksungleichung (kurz: ∆-Ungleichung) d(i, j) ≤ d(i, k) + d(k, j); +siehe auch folgendes Bild: + k + b + + + + + i b b + j +Bemerkung 7.3. Das TSP bleibt NP-vollständig im symmetrischen Fall mit Dreiecks- +ungleichung. +Definition 7.4. Es sei G ein Multigraph mit parallelen Kanten. Ein Eulerscher Kreis +ist ein Kreis, der jede Kante von G genau einmal besucht. +Beispiel 7.5. (Königsberger Brückenproblem) + + A + + B C + + D +Frage: Gibt es einen Rundweg, bei dem man über jede der 7 Brücken genau einmal +läuft? + Wir können das Brückenproblem in ein Graphproblem wie folgt transformieren: + + + A + + B C + + D + + 227 + 7 Approximative Algorithmen + + + Frage: Gibt es einen Eulerschen Kreis in dem Graphen? + Die Frage nach einem Eulerschen Kreis lässt sich durch das folgende Kriterium beant- +worten. + +Satz 7.6. Es sei G ein zusammenhängender Multigraph. Dann sind äquivalent: + + (a) G ist Eulersch (d.h. besitzt einen Eulerschen Kreis). + + (b) Jeder Knoten in G hat geraden Grad. + + (c) Die Kantenmenge von G kann in disjunkte Kreise zerlegt werden. + +Beweis. zur Übung. + + Mit Hilfe von dem obigen Kriterium sieht man nun, dass der obige Graph zum +Königsberger Brückenproblem Knoten mit ungeradem Grad besitzt und es daher keinen +entsprechenden Rundweg gibt. + Wir betrachten nun den folgenden Algorithmus: + +Algorithmus ∆T SP1 (V = {1,...,n}, D = (d(i,j)) ) + 1 Berechne einen minimalen spannenden Baum T des Graphen + 2 Kn mit Distanzen d(i, j); + 3 Konstruiere einen Multigraphen G aus T durch Verdoppeln + 4 aller Kanten; + 5 Bestimme einen Eulerschen Kreis K in G; + 6 Bestimme eine Rundreise R aus K durch Abkürzungen; + 7 return Rundreise R. + +Beispiel 7.7. Betrachte den Graphen mit V = {A, B, C, D, E}; siehe folgendes Bild +(links). + + B + b + B + b + + + 1 2 1 + + C b 1 b A C b 1 b A + 2 3 + + 2 3 3 2 2 2 + + b b b b + + D 2 E D E + + Eine optimale Tour ist [C, B, D, E, A, C] der Länge 8. Der Algorithmus berechnet einen +Baum T mit Gewicht w(T ) = 6; siehe obiges Bild (rechts). Wir erhalten den Graphen +G durch Verdopplung der Kanten mit w(G) = 12; siehe folgendes Bild: + + + +228 + 7.1 Traveling Salesman Problem + + B + b + B + b + + 1 1 + 1 1 + C b 1 b A C b 1 b A + 1 1 + 2 2 2 3 2 + 2 2 2 + 2 2 + + b b b b + + D E D E + + Als Eulerschen Kreis erhalten wir K = [C, A, E, A, C, B, C, D, C] mit Länge d(K) = +12; und durch die Abkürzungen (d.h. Überspringen der schon vorher besuchten Knoten) +erhalten wir die Rundreise R = [C, A, E, B, D, C] mit Länge d(R) = 10. + +Satz 7.8. Der Algorithmus ∆T SP1 hat eine multiplikative Güte 2; d.h. d(R) ≤ 2OP T (I), +wobei OP T (I) die Länge einer minimalen Rundreise zu der Eingabe I mit V = {1, . . . , n} +und Distanzen D = (d(i, j)) ist. + +Beweis. Die Behauptung folgt aus den folgenden Ungleichungen: + + (1) Das Gewicht w(T ) ≤ OP T (I), da eine optimale Rundreise nach Weglassen ei- + ner Kante einen spannenden Baum ergibt und w(T ) das kleinste Gewicht eines + spannenden Baums ist. + + (2) d(K) = 2w(T ) ≤ 2OP T (I). Die erste Gleichung folgt wegen der Verdopplung der + Kanten und die Ungleichung wegen (1). + + (3) d(R) ≤ d(K). Das gilt, da wegen der ∆-Ungleichung die folgende Ungleichung gilt: + d(i, j) ≤ d(i, `1 ) + d(`1 , `2 ) + . . . + d(`t , j); siehe auch folgendes Bild: + + `1 + b + `t + b + + + + + b b + + i j + +Formal zeigt man die obige Ungleichung per Induktion nach t ≥ 1. + +Bemerkung 7.9. Es gibt Beispiele für Eingaben I zu dem Traveling Salesman Problem +mit OP T (I) = n, bei denen der Algorithmus ∆T SP1 eine Rundreise der Länge 2n − 2 +berechnet. + +Beweis. Zur Beweisidee geben wir hier ein Beispiel mit n = 8 Knoten. Betrachte ei- +nen Stern mit Knoten 8 in der Mitte und Knoten 1, . . . , 7 außen, die über einen Kreis +miteinander verbunden sind. Formal haben wir V = {1, . . . , 8} und E = {{i, 8}|i = +1, . . . , 7} ∪ {{i, i + 1}|i = 1, . . . , 6} ∪ {{7, 1}}. Alle Kanten haben die Distanz 1; die +anderen Nicht-Kanten haben Distanz 2. + + + + 229 + 7 Approximative Algorithmen + + + Eine optimale TSP Tour hat die Form [1, 2, 3, 4, 5, 6, 7, 8, 1] der Länge 8. Dagegen +kann der MST Algorithmus einen Stern mit 8 als Mittelpunkt und allen anderen Knoten +außen generieren; d.h. T = (V, E 0 ) hat die Kantenmenge E 0 = {{i, 8}|i = 1, . . . , 7}. +Durch Verdopplung der Kanten erhalten wir einen Multigraphen mit 2n − 2 Kanten. +Eine mögliche Eulertour ist + + [8, 1, 8, 3, 8, 5, 8, 7, 8, 2, 8, 4, 8, 6, 8]. + +Der Algorithmus erzeugt dann über die Abkürzungen die folgende Tour: + + [8, 1, 3, 5, 7, 2, 4, 6, 8] + +mit Länge 14 = 2n − 2. + +Definition 7.10. Ein Matching in einem ungerichteten Graphen G = (V, E) ist ei- +ne Teilmenge E 0 ⊆ E von Kanten, von denen keine zwei Kanten einen gemeinsa- +men +P Endpunkt gemeinsam haben. 0Bei Gewichten d(i, j) auf den Kanten sei d(E ) = + 0 + + {i,j}∈E 0 d(i, j) das Gewicht von E . Ein Matching ist perfekt, falls jeder Knoten v ∈ V +dabei überdeckt wird; d.h. 2|E 0 | = |V |. + +Beispiel 7.11. Ein bipartiter Graph G und ein perfektes Matching: + b b + + + + + b b + + + + + b b + + + + +Satz 7.12. (Lawler 1976): Ein perfektes Matching in einem Graphen G = (V, E) mit +minimalem Gewicht kann in Zeit O(|V |3 ) berechnet werden. + +Beweis. siehe z.B. Vorlesung Discrete Optimization. + + Eine bessere Approximationsgüte erhalten wir, in dem wir die Knoten mit ungeraden +Grad optimaler durch Kanten augmentieren. Hier hilft uns ein Matching mit minimalem +Gewicht. Man beachte, dass die Anzahl der Knoten mit ungeradem Grad gerade ist; +siehe auch Lemma 5.4. + +Algorithmus ∆T SP2 (V = {1,...,n}, D = (d(i,j)) ) + 1 Berechne einen minimalen spannenden Baum T des + 2 Graphen Kn mit Gewichten d(i, j); + 3 Bestimme die Menge X der Knoten in T, + 4 die ungeraden Grad haben; + 5 Bilde den vollständigen Graphen H auf X + + + +230 + 7.1 Traveling Salesman Problem + + + 6 mit Gewichten d(i, j) für i, j ∈ X mit i 6= j. + 7 Bestimme ein perfektes Matching K in H mit minimalem + 8 Gewicht. + 9 Bilde den Multigraphen G, der aus T durch +10 Hinzufügen aller Kanten aus K entsteht. +11 Bestimme einen Eulerschen Kreis C in G; +12 Bestimme eine Rundreise R aus C durch Abkürzungen; +13 return Rundreise R. + +Satz 7.13. (Christofides): Der Algorithmus ∆T SP2 hat eine multiplikative Güte 1.5; +d.h. d(R) ≤ 1.5OP T (I). + +Beweis. Es gilt d(C) = w(T ) + d(K) ≤ OP T (I) + d(K). Man zeige nun, dass d(K) ≤ +OP T (I)/2. Daraus folgt dann die Behauptung. + Sei dazu (i1 , . . . , i2m ) mit |X| = 2m die Reihenfolge, in der die Knoten von X in einer +optimalen Tour δ von I durchlaufen werden. Betrachte die Matchings: + + M1 = {{i1 , i2 }, {i3 , i4 }, . . . , {i2m−1 , i2m }} + M2 = {{i2 , i3 }, {i4 , i5 }, . . . , {i2m , i1 }} + + Betrachte eine optimale Tour und die Knoten aus X: + + x + b b + y + i1 b b + i2 + b b + + + + + b b + + + + + i6 b b i3 + b + + + b + + b + + + b b + + i5 i4 + b + + + + + Wegen der ∆-Ungleichung gilt: OP T (I) ≥ d(i1 , i2 ) + d(i2 , i3 ) + . . . + d(i2m , i1 ) = +d(M1 ) + d(M2 ). Daher gilt: OP T (I) ≥ d(M1 ) + d(M2 ) ≥ 2d(K), da K ein perfektes +Matching bezüglich X mit minimalem Gewicht ist. Deswegen gilt: d(K) ≤ OP T (I)/2 +und damit die Behauptung. + +Beispiel 7.14. Zu unserem Graphen im obigen Beispiel erhalten wir den folgenden +induzierten Graphen H rechts mit einem perfektem Matching K und Gewicht w(K) = 3: + + + + 231 + 7 Approximative Algorithmen + + Bb + + + 1 2 H + B b + 3 b + E + C b 1 b A + 2 3 + 2 3 + 1 2 + 2 3 3 2 K + MST b b + + C 2 D + b b + + D 2 E + + Als Multigraphen G erhalten wir dann + B + b + B + b + + 1 1 2 + 1 1 + C b + 1 b A C b + 1 b A + + + 2 2 2 2 + + + b + 2 b b + 2 b + + D E D E + + Ein Eulerkreis in G ist C = [C, B, C, A, E, D, C] und nach den Abkürzungen erhalten +wir R = [C, B, A, E, D, C] mit Länge d(R) = 9. +Lemma 7.15. Es gibt Beispiele für Eingaben I zu dem Traveling Salesman Problem mit +OP T (I) = n, bei denen der Algorithmus ∆T SP2 von Christofides eine Rundreise der +Länge (n − 1) + bn/2c berechnet. +Beweis. zur Übung. + + +7.2 Rucksackproblem +Dass das Rucksackproblem N P -vollständig ist, haben wir schon am Ende des letzten Ka- +pitels gesehen. Hier wollen wir nun einige einfache Approximationsalgorithmen studieren. + +Gegeben: + • n Gegenstände mit Gewichten w0 , . . . , wn−1 ∈ N und Gewinnen p0 , . . . , pn−1 ∈ N. + + • Rucksack der Kapazität B ∈ N. +Gesucht: Teilmenge I ⊆ {0, . . . , n − 1} mit i∈I wi ≤ B und + P + + X X X + pi = max{ pi |I 0 ⊆ {0, . . . , n − 1}, wi ≤ B}. + i∈I i∈I 0 i∈I 0 + + + + +232 + 7.2 Rucksackproblem + + +Satz 7.16. Das Entscheidungsproblem zum Rucksackproblem ist NP-vollständig; sogar +wenn wi = pi für alle i = 0, . . . , n − 1. + +Beweis. siehe Kapitel Einführung in die Komplexitätstheorie. + + Zunächst betrachten wir einen einfachen Greedy Algorithmus (GA): + +Algorithmus GA(I = ((w0 , p0 ), . . . , (wn−1 , pn−1 ), B)) + 1 sortiere die Gegenstände, so dass p0 /w0 ≥ p1 /w1 ≥ . . . ≥ pn−1 /wn−1 ; + 2 setze S = ∅; + 3 for i = + P0 to n − 1 do + 4 if ( j∈S wj ) + wi ≤ B then + 5 S = S ∪ {i} + 6 fi + 7 od + 8 return(S). + + Es sei GA(I) der Gesamtgewinn der durch Algorithmus GA berechneten Lösung S ⊆ +{0, . . . , n − 1} und OP T (I) der Gesamtgewinn einer optimalen Lösung zur Eingabe I. + +Satz 7.17. (a) Die Laufzeit des Algorithmus GA ist O(n log n). + + (b) Der Algorithmus GA hat eine multiplikative Güte größer gleich B−1 (d.h. OP T (I)/GA(I) ≥ + (B − 1)). + Der Algorithmus GA hat daher keine konstante Worst-Case Güte. + +Beweis. Zu (a): hierzu speichere bei jedem Hinzufügen eines Gegenstands die momentan +benutze Kapazität K. Dann geht Schritt (2 − 8) in O(n) Zeit. Die Gesamtlaufzeit ergibt +sich durch das Sortieren in O(n log n). + Zu (b): Betrachte das folgende Beispiel: I = ((w0 , p0 ) = (1, 1), (w1 , p1 ) = (B, B − +1), B). Hier gilt p0 /w0 = 1 und p1 /w1 = (B − 1)/B < 1. Der Algorithmus liefert hier +S = {0} mit Gewinn 1; optimal ist aber Sopt = {1} mit Gewinn B − 1. + In diesem Fall gilt OP T (I)/GA(I) = B − 1. D.h. im allgemeinen Fall ist die Rate +unbeschränkt. + +Hinweis: Man beachte, dass GA in dem obigen Beispiel das Element mit maximalem +Profit pmax ignoriert hat. Aus diesem Grunde betrachten wir eine leichte Modifikation +von GA, die sofort hilft die Approximationsgüte zu beschränken. + +Algorithmus MGA(I = ((w0 , p0 ), . . . , (wn−1 , pn−1 ), B)) + 1 berechne Lösung S1 mit Algorithmus GA, + 2 berechne Lösung S2 = {j} mit pj = maxi∈{0,...,n−1} pi , + 3 wähle Lösung S ∈ {S1 , S2 } mit größerem Gewinn. + 4 return(S). + + + + 233 + 7 Approximative Algorithmen + + + Es sei M GA(I) der Gesamtgewinn der durch Algorithmus M GA berechneten Lösung +S ⊆ {0, . . . , n − 1} und OP T (I) der Gesamtgewinn einer optimalen Lösung zur Eingabe +I. + +Satz 7.18. Die Worst-Case Güte von Algorithmus M GA ist 2 (d.h. OP T (I)/M GA(I) ≤ +2 für alle Eingaben I). + +Beweis. Betrachte relaxierte Version des Rucksackproblems, bei dem Gegenstände frak- +tional genommen werden dürfen. Es sei OP Tf (I) der maximale Wert einer fraktionalen +Lösung x∗ . Es gilt dann OP T (I) ≤ OP Tf (I), da der Lösungsraum sich vergrößert und +ein Max-Problem vorliegt. + Eine fraktionale Lösung x∗ von max nj=1 xj pj mit j xj wj ≤ B und xj ∈ [0, 1] für + P P +alle j = 1, . . . , n nimmt Gegenstände in der sortierten Reihenfolge + + p1 /w1 ≥ p2 /w2 ≥ . . . ≥ pn /wn , + +wobei die ersten k Gegenstände ganzzahlig (d.h. x∗j = 1) und der letzte Gegenstand +fraktional mit x∗k+1 ∈ [0, 1) gewählt wird. +Beachte: Es gilt kj=1 wj ≤ B und k+1 j=1 wj > B. Insgesamt haben wir + P P Pk+1 ∗ + j=1 xj wj = B. +Falls wir einen fraktionalen Gegenstand haben mit x∗k+1 > 0 ist, so gilt kj=1 wj < B. + P + + Im Folgenden zeigen wir, dass eine optimale fraktionalen Lösung die obige Form hat. +Angenommen, wir haben ein Teilstück eines Gegenstands ` > k + 1 der Länge x∗` w` im +Rucksack mit p` /w` < pk+1 /wk+1 und x∗` > 0. Dann muss ein anderes Stück (1 − x∗i )wi +mit i ≤ k +1 und x∗i < 1 im Rucksack fehlen (wegen den obigen Bedingungen). Betrachte +die minimale Länge length = min{(1 − x∗i )wi , x∗` w` }. Das folgende Bild veranschaulicht +die genaue Situation: + pj + wj (1 − x∗i )wi x∗` w` Rucksack: grün + + y` w` + + ... ... ... + ... + + 1 2 i k ` n wj + + austauschen + Dann gilt length = yi wi = y` w` mit yi ∈ (0, 1 − x∗i ] und y` ∈ (0, x∗` ] und pi /wi ≥ +pk+1 /wk+1 > p` /w` . +Frage: Wie bestimmt man yi , y` ? + Man beachte, dass length = min{(1 − x∗i )wi , x∗` w` } > 0. + +Fall 1: (1 − x∗i )wi = length. Hier setze yi = (1 − x∗i ) und y` = length/w` . + +Fall 2: x∗` w` = length. Hier setze y` = x∗` und yi = length/wi . + + + +234 + 7.2 Rucksackproblem + + + Ersetze nun die Länge y` w` in der Lösung durch yi wi . Der Profitgewinn durch diesen +Austausch ist dann yi pi − y` p` und es gilt: + + yi pi − y` p` > yi (p` /w` )wi − y` p` + = yi wi p` /w` − y` p` + = y` w` p` /w` − y` p` + = y` p` − y` p` = 0. + +D.h. der Profitgewinn ist echt positiv und damit kann ein solcher obiger Fall nicht +auftreten. + Nun kommen wir zur Abschätzung der Approximationsgüte: Da k + 1 das erste Item +ist, dass der Algorithmus GA nicht nimmt, gilt + + OP Tf (I) ≤ GA(I) + pk+1 . + +Der Algorithmus M GA berechnet Lösung mit Wert GA(I) oder pmax ≥ pk+1 (bzw. sogar +genauer mit Wert max{GA(I), pmax }). Daher gilt + + OP T (I) ≤ OP Tf (I) ≤ GA(I) + pk+1 + ≤ GA(I) + pmax ≤ 2 max{GA(I), pmax } + = 2 · M GA(I) + + + +Bemerkung 7.19. (a) Der Algorithmus M GA läuft in Zeit O(n log n). + + (b) Durch eine Modifikation von M GA können wir auch einen approximativen Algo- + rithmus angeben mit Worst Case Güte 2, der in O(n) Zeit läuft. + +Beweis. Wir verwenden als Idee zur Laufzeitverbesserung (b) hierzu den Median-Algorithmus +um das sogenannte Split-Item k + 1 in O(n) Zeit zu finden. + + Der folgende Algorithmus verallgemeinert die Idee vom MGA Algorithmus, in dem +er alle k-elementigen Teilmengen durchprobiert, in den Rucksack vorplaziert, wenn das +möglich ist, und mit Hilfe vom GA Algorithmus die Restkapazität auffüllt. +Algorithmus von Sahni + +Gegeben: Parameter k ∈ {0, . . . , n}. +Idee: Für jedes k definiere Algorithmus Ak wie folgt: + + (1) wähle eine Teilmenge S mit |S| ≤ k Gegenständen, die am Anfang im Rucksack + liegen, + + (2) wende Algorithmus GA auf die übrigen Elemente an und füge die ausgewählten + Gegenstände zusätzlich in den Rucksack ein. + + + + 235 + 7 Approximative Algorithmen + + +Wende (1) + (2) auf alle möglichen Teilmengen S mit |S| ≤ k an und wähle Lösung mit +maximalem Gewinn. + Die Technik nennt man k-Enumeration. Es sei Ak (I) der Gewinn der Lösung, den +der Algorithmus Ak von Sahni bezüglich einer Eingabe I und Parameter k ∈ {0, . . . , n} +produziert. + +Satz 7.20. Für alle k ≥ 1 gilt: Ak hat Güte ≤ 1 + 1/k und Laufzeit TAk (n) = O(nk+1 ). + +Beweis. siehe Vorlesung Effiziente Algorithmus. + + Idee für ein schnelleres Verfahren: Konstruiere aus dem exakten Algorithmus +Knapsack aus Kapitel 2 zum Rucksackproblem einen polynomiellen approximativen +Algorithmus. + Erinnerung: Das dynamische Programm zum Rucksackproblem hat eine nicht-polynomielle +(bzw. eine sogenannte pseudo-polynomielle) Laufzeit O(n2 ·pmax ), wobei pmax = max{pi |0 ≤ +i ≤ n − 1}. Hierbei könnte pmax exponentiell in der Eingabegröße sein; siehe auch Be- +merkung 2.20. + Idee zu Algorithmus AK + + (1) konstruiere eine Eingabe IK mit Gewinnen p0i = bpi /Kc (alle anderen Größen wi + und B bleiben gleich). + + (2) wende Algorithmus Knapsack auf Eingabe IK an. + + Der Algorithmus AK berechnet eine Lösung S ⊆ {0, . . . , n−1} mit Größe + P + i∈S wi ≤ B +und maximalem Gewinn i∈S p0i . + P + Wir entwickeln aus AK ein Verfahren A durch die Wahl + pmax + K= . + (1/ + 1)n + +Satz 7.21. Der Algorithmus A läuft in Zeit O(n3 /) und hat eine Worst-Case Güte +OP T (I)/A (I) ≤ 1 + . + + Man nennt eine solche Familie von Algorithmen (A ) ein vollständiges polynomielles +Approximationsschema (FPTAS). + +Beweis. (a) zur Laufzeit: Für die skalierten Profite gilt: p0max = maxi p0i = maxi bpi /Kc = +maxi bpi /pmax (1/ + 1)nc = b(1/ + 1)nc = O(n/). + Der Algorithmus Knapsack angewandt auf die skalierte Instanz hat dann eine Laufzeit +von + O(n2 p0max ) = O(n2 n/) = O(n3 /). + (b) zur Güte: Es gilt zunächst + + OP T (I) ≤ K · OP T (IK ) + K · n (1) + + Da p0i = bpi /Kc ≥ pi /K − 1, haben wir pi ≤ K · (p0i + 1) und deswegen haben wir + + + +236 + 7.2 Rucksackproblem + + + + pi ≤ i∈Iopt (Kp0i + K) + P P + OP T (I) = i∈Iopt + 0 + P + ≤ i∈Iopt Kpi + K · n ≤ K · OP T (IK ) + K · n + + Beachte, dass Iopt eine zulässige Lösung von IK ist und (IK )opt i.a. ein höheren Gewinn +als Iopt bezüglich Instanz IK hat. Daneben gilt: + + K · OP T (IK ) ≤ AK (I) (2) + Da p0i ≤ pi /K gilt + + K · OP T (IK ) = K · i∈(IK )opt p0i ≤ K · i∈(IK )opt pi /K + P P + P + = i∈(IK )opt pi ≤ AK (I). + + Aus (1) + (2) folgt + OP T (I) ≤ AK (I) + K · n. +Da alle wi ≤ B (ansonsten kann Gegenstand i aus der Instanz gelöscht werden), gilt +OP T (I) ≥ pmax = maxi pi . + Wir erhalten + OP T (I) AK (I) + Kn + ≤ + A (I) AK (I) + Kn + =1+ + AK (I) + Kn + ≤1+ + OP T (I) − Kn + Kn + ≤1+ + pmax − Kn + = 1 + . + +Die letzte Gleichung gilt, da + pmax + Kn (1/+1)n · n + = pmax + pmax − Kn pmax − (1/+1)n n + 1 1 + (1/+1) 1/+1 + = 1 = 1/+1−1 + 1 − (1/+1) + 1/+1 + 1 + = = . + 1/ + +Weitere Approximationsschema + + • O(n2 /) (Ibarra und Kim 1975). + + • O(n log(1/) + 1/4 ) (Lawler 1979). + + + + 237 + 7 Approximative Algorithmen + + + • O(n min{log n, log(1/)} + 1/2 log(1/) min{n, 1/ log(1/)}) (Kellerer, Pferschy + 2004). + + • Õ(n + (1/)5/2 ) (Rhee 2015). + + • Õ(n + (1/)12/5 ) (Chan 2018). + + • Õ(n + (1/)9/4 ) (Jin 2019). + +In der Õ Notation vernachlässigt man log(.)-Terme in der Laufzeit. + +Bemerkung 7.22. Es gibt kein FPTAS mit einer Laufzeit O((n + 1/)2−δ ), außer +(min, +) Convolution hat einen Algorithmus mit einer subquadratischen Laufzeit (Cygan +u.a. 2017 bzw. Künnemann u.a. 2017). + + +7.3 Scheduling +Der Begriff Scheduling bescheibt die Problematik der Erstellung eines Ablaufplans bzw. +Schedules, der Prozessen zeitlich begrenzt Ressourcen, wie etwa Maschinen bzw. Prozes- +soren oder Speicher, zuteilt. Dabei untersucht man die Optimierung bzw. Approximation +diverser Zielfunktionen. Eine klassische Variante wird formal als P ||Cmax angegeben und +lässt sich wie folgt formulieren. +Gegeben: + n Jobs J = {J1 , . . . , Jn } mit Ausführungszeiten p1 , . . . , pn ∈ N und m identische +Maschinen. +Gesucht: + Partition von J in m Teilmengen B1 , . . . , Bm mit minimaler maximaler Last + X + Cmax = max pj . + 1≤i≤m + Jj ∈Bi + + +Den Wert Cmax nennt man auch den Makespan des Schedules. Das Problem wird be- +zeichnet durch P ||Cmax bzw. P m||Cmax , wenn die Anzahl der Maschinen m konstant ist. +Im Folgenden betrachten wir eine Instanz für das Schedulingproblem P ||Cmax : + + Tab. 7.1: Schedulinginstanz + + Jj J1 J2 J3 J4 J5 + m=2 + pj 1 2 1 3 1 + + Für diese Instanz mit zwei Maschinen können wir etwa die beiden folgenden Ablauf- +pläne/Schedules als zulässig erkennen: + + • B1 = {J1 , J2 , J5 }, B2 = {J4 , J5 } mit Makespan 4. + + + +238 + 7.3 Scheduling + + + M1 J1 J2 J3 + M2 J4 J5 + Zeit + 0 1 2 3 4 5 + + + • B1 = {J1 , J3 , J4 }, B2 = {J2 , J5 } mit Makespan 5. + + M1 J1 J3 J4 + M2 J2 J5 + Zeit + 0 1 2 3 4 5 + + + Man erkennt leicht, dass der erste Ablaufplan ein optimaler Ablaufplan ist. Der op- +timale Makespan ist also 4. Wie schwer ist nun dieses Problem? Der folgende Satz gibt +uns Auskunft. + +Satz 7.23. Das Schedulingproblem auf identischen Maschinen P m||Cmax ist NP-vollständig +sogar für m = 2 Maschinen. + +Beweis. Dies folgt durch eine Reduktion des Partitionsproblems. + + Man mache sich klar, dass daraus unmittelbar auch die NP-Vollständigkeit von P ||Cmax +folgt. Wir wollen nun einen ersten Approximationsalgorithmus studieren, den wir List- +Scheduling nennen. + +Algorithmus ListScheduling(L=(J1 , . . . , Jn ),m) + 1 for i=1 to m do + 2 Ei = 0; Bi = ∅; + 3 od + 4 for j=1 to n do + 5 wähle Job Jj aus Liste L; + 6 wähle Maschine Mi mit minimaler Last Ei ; + 7 Bi = Bi ∪ {Jj }; + 8 Ei = Ei + pj ; + 9 od + + Es sei LS(I) die Länge eines List Schedules und OP T (I) die Länge eines optimalen +Schedules zur Eingabe I. + +Satz 7.24. (a) Für alle Eingaben I = (L, m) gilt LS(I)/OP T (I) ≤ 2 − 1/m. + + (b) Es existiert eine Eingabe I ∗ mit LS(I ∗ ) = (2 − 1/m)OP T (I ∗ ). Der Algorithmus + hat also eine multiplikative Güte bzw. absolute Worst Case Güte von 2 − 1/m. + + + + 239 + 7 Approximative Algorithmen + + +Beweis. Zu (a): O.B.d.A. hat Maschine M1 nach der Zuordnung die höchste Last L = + j∈B1 pj (ansonsten nummeriere die Maschinen um). Es sei Jk der letzte Job, der auf +P +Maschine M1 fertig wird. Dann haben alle Maschinen eine Last Li ≥ L − pk . Zu dem +Zeitpunkt, wo Jk der Maschine M1 zugeordnet wurde, hatte M1 die kleinste Last L − pk . +Wir illustieren dies in Abb. 7.1. + + M1 Jk + M2 + M3 + .. + . + Mm + Zeit + 0 L − pk L + + Abb. 7.1: Maschine M1 hat mit Jk bzw. ohne Jk die größte Last bzw. kleinste Last + + Daraus folgt sofort die folgende Abschätzung: + n + X + pi ≥ m(L − pk ) + pk . + i=1 + Daneben gilt (da die gesamte Last Ppi auf m Maschinen verteilt werden muss und + P +mindestens eine Maschine eine Last ≥ pi /m hat): + n + X + pi /m ≤ OP T (I). + i=1 + +Da nun LS(I) = L ist, gilt + n + X + OP T (I) ≥ pi /m ≥ m(L − pk )/m + pk /m + i=1 + = L − (1 − 1/m)pk = LS(I) − (1 − 1/m)pk + +Da OP T (I) ≥ pk folgt nun OP T (I) ≥ LS(I) − (1 − 1/m)OP T (I) bzw. + + LS(I) ≤ (2 − 1/m)OP T (I). + +Zu (b): zur Übung. + + Wir haben gesehen, dass ein erstaunlich einfacher Algorithmus schon eine Güte von 2 +garantieren kann. Allerdings fällt auf, dass die beliebige Wahl des nächsten Jobs (in Zeile +5) mitunter schlecht sein kann. Es scheint besser zu sein, die Jobs absteigend ihrer Aus- +führungszeit nach zu plazieren, was uns zum folgenden Algorithmus LPT Scheduling +führt. + + + +240 + 7.3 Scheduling + + +Algorithmus LPT Scheduling(I=(J,,m)) + 1 sortiere die Jobs in J so, dass p1 ≥ p2 . . . ≥ pn ; + 2 wende den List Scheduling Algorithmus an auf I=(L,m) mit L=(J1 , . . . , Jn ) + + Es sei LP T (I) die Länge eines LPT Schedules bezüglich einer Eingabe I = (J, m). + +Satz 7.25. Der LPT Algorithmus hat eine absolute Worst Case Rate von 4/3 − 1/(3m); +d.h. + LP T (I) ≤ (4/3 − 1/(3m))OP T (I). (∗) + +Beweis. Annahme: Es existiert eine Jobmenge J und Maschinenzahl m, die die Be- +hauptung (∗) nicht erfüllt. Der Satz gilt sofort für m = 1 (in diesem Fall gilt nämlich +LP T (I) = OP T (I)). Daher nehmen wir an: m ≥ 2 und n ist minimal (d.h. wir haben +ein minimales Gegenbeispiel). +Annahme: Es existiert ein Job Jr mit r < n, für den die Fertigstellungszeit fr = +LP T (I = (J, m)) gilt. Betrachte J 0 = {J1 , . . . , Jr } mit Liste L0 = (J1 , . . . , Jr ). Dann gilt +LP T (I = (J, m)) = LP T (I 0 = (J 0 , m)) und der optimale Wert OP T (I 0 ) ≤ OP T (I). +Damit haben wir + LP T (I 0 ) LP T (I) + ≥ > 4/3 − 1/(3m) + OP T (I 0 ) OP T (I) +ein kleineres Gegenbeispiel gefunden. Widerspruch zur Minimalität von n. + Es gilt also fk < LP T (I = (J, m)) für alle k < n. Desweiteren haben wir die Unglei- +chungen: + n + 1 X + OP T (I) ≥ pi (1) + m + i=1 + n−1 + (2) + X + pi ≥ msn = m(LP T (I) − pn ), + i=1 + +wobei sn der Startzeitpunkt von Job Jn ist. + + pn + Jn + + m Jobs J1 , . . . , Jn−1 + + + Zeit + 0 sn LPT(J) + + + Beachte: Alle Maschinen sind bis zum Zeitpunkt sn voll beschäftigt (d.h. keine Ma- + + + + 241 + 7 Approximative Algorithmen + + +schine ist idle vor dem Zeitpunkt sn ). + + LP T (I) sn + pn + = + OP T (I) OP T (I) + n−1 + pn 1 X + ≤(2) + pi + OP T (I) mOP T (I) + i=1 + n + (m − 1)pn 1 X + = + pi + mOP T (I) mOP T (I) + i=1 + (m − 1)pn + ≤(1) + 1. + mOP T (I) + +Da (∗) für I nicht gilt, folgt + + (m − 1)pn LP T (I) + 1+ ≥ > 4/3 − 1/(3m). + mOP T (I) OP T (I) + +Daraus folgt + (m − 1)pn m−1 + > 1/3 − 1/(3m) = + mOP T (I) 3m +bzw. + OP T (I) + pn > . + 3 +(d.h. der kleinste Job Jn ist größer als OP T (I)/3). Daraus folgt, dass keine Maschine in +einem optimalen Schedule mehr als 2 Jobs ausführen kann. + Als nächstes transformieren wir einen optimalen Schedule Dopt für solch eine Jobmenge +mit pn > OP T (I)/3 in einen Schedule D̄opt , der zusätzliche Eigenschaften erfüllt. +Transformation I.1 für Dopt . Gegeben sei ein Schedule Dopt mit Makespan Cmax und +zwei Jobs Ji und Ji0 auf einer Maschine und zwei Jobs Jj und Jj 0 auf einer anderen +Maschine, wobei pi > pj und pi0 > pj 0 ist. + + + + + Ji Ji0 + + + Jj Jj 0 + + + + Zeit + 0 + +Wenn wir Ji0 und Jj 0 vertauschen, so erhalten wir den folgenden Schedule Dopt + 0 mit +Makespan Cmax ≤ Cmax : + 0 + + + + +242 + 7.3 Scheduling + + + + + Ji Jj 0 + + + Jj Ji0 + + + + Zeit + 0 + +Transformation I.2 für Dopt . Gegeben sei ein Schedule Dopt mit Makespan Cmax und +zwei Jobs Ji und Ji0 auf einer Maschine und ein Job Jj auf einer anderen Maschine, +wobei pi > pj ist. + Wenn wir den Job Ji0 auf die andere Maschine verschieben, so erhalten wir einen +Schedule Dopt + 0 ebenso mit Makespan Cmax + 0 ≤ Cmax : + + + + + Ji Ji0 + + verschieben + Jj + + + + Zeit + 0 + +Transformation II für Dopt . Gegeben sei ein Schedule Dopt mit Makespan Cmax und +zwei Jobs Ji und Ji0 auf einer Maschine mit pi < pj . In diesem Fall vertauschen wir die +Reihenfolge beider Jobs auf der entsprechenden Maschine: + + + + + austauschen + Ji Jj + + + + Zeit + 0 + +Beachte: Keine der Transformationstypen I.1, I.2 oder II erhöht den Makespan. + + + + 243 + 7 Approximative Algorithmen + + +Definition 7.26 (Lastfunktion). Bei gegebenen Lastwerten Li auf den Maschinen Mi +in einem Schedule D ist die Lastfunktion L(D) = 1≤i pj , so +folgt + + pj 0 ≥ pi0 , pi ≤ pk , pi ≥ pi0 , (∗∗) + pj ≤ pk , pj ≥ pj 0 . + + Durch Umordnen der Maschinen in D∗ erhalten wir einen Schedule D̄opt : + + + +244 + 7.3 Scheduling + + + Jk1 + Jk2 + + + + + Jks + Ji1 Ji01 + Ji2 Ji02 + + + + + Jit Ji0t + Zeit + 0 + + Hierbei gelten die Ungleichungen pk1 ≥ . . . ≥ pks sowie pi1 ≥ . . . ≥ pit und wegen (∗∗) +haben wir p0i1 ≤ . . . ≤ p0it und pks ≥ pi1 sowie pit ≥ p0it . Damit haben wir die folgende +Ordnung der Jobs: + pk1 ≥ . . . ≥ pks ≥ pi1 ≥ . . . ≥ pit ≥ p0it ≥ . . . p0i1 . + Der Schedule D̄opt ist nun äquivalent zu einem List Schedule DL bei gegebener Lis- +te L = (J1 , . . . , Jn ) mit p1 ≥ . . . ≥ pn (bis auf Vertauschen von Jobs mit gleicher +Ausführungszeit). Daraus folgt aber + + LP T (I = (J, m)) = OP T (I = (J, m)). + Wegen LP T (I)/OP T (I) > 4/3 − 1/(3m) erhalten wir nun aber einen Widerspruch +(d.h. es kann kein Gegenbeispiel für die Ungleichung (∗) existieren). + +Bemerkung 7.28. Für jede Anzahl m von Maschinen existiert eine Eingabe Im + ∗ = + +(J ∗ , m) mit + ∗) + LP T (Im 4 1 + ∗) + = − . + OP T (Im 3 3m +Beweis. zur Übung. + +Satz 7.29. Für jede Genauigkeit  > 0 gibt es einen approximativen Algorithmus A für +P ||Cmax mit A (I) ≤ (1 + )OP T (I) und Laufzeit 2O(1/ log (1/)) + O(n). + 2 + + + +Bemerkung 7.30. Eine Familie (A ) von solchen Algorithmen nennt man ein (effizi- +entes) polynomielles Approximationsschema (EPTAS). +Details: siehe Vorlesung Effiziente Algorithmen. + + + + 245 + 7 Approximative Algorithmen + + +Übungsaufgaben +Übung 7.31. Zeigen Sie: + + • Ein zusammenhängender ungerichteter Graph G = (V, E) enthält eine Eulertour + genau dann, wenn jeder Knoten in V geraden Knotengrad hat. + + • Ein stark zusammenhängender gerichteter Graph G = (V, E) enthält eine Eulertour + genau dann, wenn für jeden Knoten v ∈ V gilt, dass din (v) = dout (v). + + + + +246 + \ No newline at end of file diff --git a/uni/aak (Copy)/alle_hausaufgaben.pdf b/uni/aak (Copy)/alle_hausaufgaben.pdf new file mode 100644 index 0000000..2bd9ce0 --- /dev/null +++ b/uni/aak (Copy)/alle_hausaufgaben.pdf @@ -0,0 +1,7959 @@ +%PDF-1.3 +% +1 0 obj +<< /Metadata 3 0 R /Pages 4 0 R /Type /Catalog >> +endobj +2 0 obj +<< /Author () /CreationDate (D:20250713031408+02'00') /Creator (TeX) /ModDate (D:20250713031408+02'00') /Producer (pdfTeX-1.40.25) >> +endobj +3 0 obj +<< /Subtype /XML /Type /Metadata /Length 793 >> +stream + + + + + + + + +endstream +endobj +4 0 obj +<< /Count 21 /Kids [ 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 10 0 R 11 0 R 12 0 R 13 0 R 14 0 R 15 0 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/Info 2 0 R /Root 1 0 R /Size 651 /ID [<58d2f731fa4fb722ecc943c307d52835><58d2f731fa4fb722ecc943c307d52835>] >> +startxref +1312234 +%%EOF diff --git a/uni/aak (Copy)/alle_hausaufgaben.txt b/uni/aak (Copy)/alle_hausaufgaben.txt new file mode 100644 index 0000000..6e3f255 --- /dev/null +++ b/uni/aak (Copy)/alle_hausaufgaben.txt @@ -0,0 +1,817 @@ + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 09.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 9 + + +Definition 1 (Vertex Cover (VC)). Für einen (ungerichteten) Graphen G = (V, E) ist ein Vertex +Cover eine Menge von Knoten C ⊆ V , sodass u ∈ C oder v ∈ C für alle { u, v } ∈ E gilt, also ist +für alle Kanten mindestens ein Endpunkt in C. +Für das V ERTEX C OVER Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gegeben und +es wird gefragt, ob ein Vertex Cover mit Größe (Anzahl der Knoten) höchstens k in G existiert. + +Definition 2 (Feedback Vertex Set (FVS)). Für einen gerichteten Graphen G = (V, E) ist ein +Feedback Vertex Set eine Menge von Knoten X ⊆ V , sodass G \ X kreisfrei ist. +Für das F EEDBACK V ERTEX S ET Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gege- +ben und es wird gefragt, ob ein Feedback Vertex Set mit Größe (Anzahl der Knoten) höchstens +k in G existiert. + + + 1 6 2 + + + 5 + 8 7 + 3 + 4 + Abbildung 1: Beispiel Graph. + +Hausaufgabe 9.1 (Nichtdeterministischer Algorithmus für VC) (5 Punkte) + 1. Finden Sie ein minimales Vertex Cover in Abbildung 1. Für ein Vertex Cover ist die + Orientierung der Kanten nicht relevant. + 2. Zeigen Sie V ERTEX C OVER ∈ NP, indem Sie einen nichtdeterministischen Algorithmus + angeben, der V ERTEX C OVER in Polynomialzeit löst. Begründen Sie die Korrektheit und + die Laufzeit von Ihrem Algorithmus. + +Lösung. 1. Ein minimales Vertex Cover ist zum Beispiel { 3, 5, 6, 7, 8 }. + 2. Unser nichtdeterministischer Algorithmus geht wie folgt vor: Für jeden Knoten wird + nichtdeterministisch entschieden, ob dieser im Vertex Cover sein soll. Nachdem dies für + alle Knoten entschieden wurde, wird überprüft, ob nicht mehr als k Knoten gewählt wur- + den und ob für jede Kante mindestens ein Endpunkt im Vertex Cover liegt. + Existiert ein Vertex Cover im Graphen, existiert auch eine Reihe von nichtdeterministi- + schen Entscheidungen, die diese Teilmenge von Knoten auswählt. Diese Teilmenge wird + dann auch von dem Algorithmus akzeptiert. Damit ist der Algorithmus korrekt. + Beide diese Schritte können naiv in Polynomialzeit erledigt werden, also löst dieser Al- + gorithmus das Vertex Cover Problem in Polynomialzeit. + + +Bewertung 1 Punkt für das Beispiel und 4 Punkte für den Algorithmus, dabei 2 Punkte für +die Beschreibung und jeweils 1 Punkt für Korrektheit und Laufzeit. + +Hausaufgabe 9.2 (Polynomieller Verifizierer für FVS) (5 Punkte) + 1. Finden Sie ein minimales Feedback Vertex Set in Abbildung 1. + 2. Zeigen Sie F EEDBACK V ERTEX S ET ∈ NP, indem Sie einen polynomiellen Verifizierer + für F EEDBACK V ERTEX S ET angeben. Begründen Sie die Korrektheit und die Laufzeit + von Ihrem Verifizierer. + +Lösung. 1. Ein minimales FVS ist zum Beispiel { 5, 6 }. + 2. Das Zertifikat hat die Form einer Teilmenge X ⊆ V , die beschreibt, welche Knoten das + Feedback Vertex Set bilden sollen. Zum Beispiel über ein Bit pro Knoten. Nun wird + überprüft, ob |X| ≤ k gilt und ob G \ X azyklisch ist. + Hat der Graph ein Feedback Vertex Set von Größe maximal k, gibt es ein Zertifikat, wel- + ches dieses beschreibt. Dieses Zertifikat wird dann von unserem Verifizierer akzeptiert, + damit ist der Verifizierer korrekt. + Die Größe von X zu überprüfen ist naiv in Polynomialzeit möglich. Für den zweiten Test + bietet sich der Algorithmus zur topologischen Sortierung aus der Vorlesung an, welcher in + Polynomialzeit eine topologische Sortierung findet, wenn der gegebene Graph azyklisch + ist. Insgesamt ist dieser Verifizierer korrekt und auch ein polynomieller Verifizierer. + + +Bewertung 1 Punkt für das Beispiel und 4 Punkte für den Verifizierer, dabei 2 Punkte für die +Beschreibung und jeweils 1 Punkt für Korrektheit und Laufzeit. + + + + +Abgabe: 16.06.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 9 + Marek Lenczewski + Matrikelnummer: 1025252 + + 22. Juni 2025 + + +Hausaufgabe 9.1 +Teil 1 +Ein minimales Vertex Cover ist C = {2, 7, 5, 4, 1}. +Die Minimale Knotenanzahl ist 5. + +Teil 2 +Der Algorithmus arbeitet wie folgt: + + • Zuerst wird für jeden Knoten v ∈ V nichtdeterministisch entschieden, ob dieser in das + Vertex Cover aufgenommen wird. Dabei wird drauf geachtet, dass die Anzahl k nicht + übersteigt. + + • Danach wird geprüft, ob für jede Kante {u, v} ∈ E entweder v ∈ C oder u ∈ C oder beides + zutrifft. + + • Falls dies für alle Kanten gilt, akzeptieren wir. Andernfalls lehnen wir ab. + + Existiert ein Vertex Cover der Größe höchstens k, dann gibt es auch eine, nichtdeterminis- +tisch ausgewählte, Knotenmenge, die dazu passt. Von dieser werden alle Kanten abgedeckt und +der Algorithmus akzeptiert. Falls es kein Vertex Cover gibt, dann lehnt der Algorithmus ab. + + Die Auswahl der Knoten und die Prüfung der Kanten läuft in polynomieller Zeit, also ist dies +ein polynomieller nichtdeterministischer Algorithmus, der VERTEX COVER löst und damit ist +V ERT EXCOV ER ∈ N P . + + +Hausaufgabe 9.2 +Teil 1 +Ein minimales Feedback Vertex Set ist X = {1, 5} + +Teil 2 +Der Verifizierer arbeitet wie folgt: + + • Es wird eine Teilmenge X ⊆ V übergeben, die das Feedback Vertex Set darstellen soll. + + • Dann wird geprüft, ob |X| ≤ k gilt und es wird abgelehnt, falls es nicht gilt. + + • Danach wird ein Graph G′ = G \ X konstruiert, in dem alle Knoten aus X und alle an X + angrenzenden Kanten entfernt wurden. + + 1 + • Auf dem Graphen G′ wird eine Tiefensuche durchgeführt, dabei werden Knoten während + der Bearbeitung grau markiert und beim Backtracking schwarz. + + • Wird eine Kante zu einem grauen Knoten entdeckt, dann wird abgelehnt, da dies einen + Kreis anzeigt. + + • Falls die Tiefensuche den gesamten Graphen durchläuft ohne abzulehnen, dann wird ak- + zeptiert. + + Existiert ein Feedback Vertex Set der Größe höchstens k, dann gibt es auch ein Zertifikat, das +genau diese Knotenmenge beschreibt. Alle Teilmengen, die die Tiefensuche überstehen, werden +von dem Verifizierer akzeptiert und sonst abgelehnt. + + Die Prüfung der übergebenen Teilmenge und die Konstruktion des Graphen laufen in polyno- +mieller Zeit. Ebenfalls die Tiefensuche läuft in polynomieller Zeit, somit ist auch der Verifizierer +polynomiell und verifiziert die Lösung für das FEEDBACK VERTEX SET in polynomieller +Zeit. Also gilt F EEDBACKV ERT EXSET ∈ N P . + + + + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 16.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 10 + + +Definition 1 (Feedback Vertex Set (FVS)). Für einen gerichteten Graphen G = (V, E) ist ein +Feedback Vertex Set eine Menge von Knoten X ⊆ V , sodass G \ X kreisfrei ist. +Für das F EEDBACK V ERTEX S ET Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gege- +ben und es wird gefragt, ob ein Feedback Vertex Set mit Größe (Anzahl der Knoten) höchstens +k in G existiert. + +Hausaufgabe 10.1 (F EEDBACK V ERTEX S ET ist NP-vollständig) (5 Punkte) +Zeigen Sie, dass F EEDBACK V ERTEX S ET NP-vollständig ist. +Hinweis: Reduzieren Sie V ERTEX C OVER auf F EEDBACK V ERTEX S ET. Wenn ein Vertex Cover +aus einem Graphen entfernt wird, hat dieser keine Kanten mehr. + +Lösung. F EEDBACK V ERTEX S ET ∈ NP ist bekannt aus der letzten Hausaufgabe. Nun zeigen +wir eine Reduktion von V ERTEX C OVER auf F EEDBACK V ERTEX S ET. Betrachte die folgende +Reduktion: +Für eine gegebene V ERTEX C OVER Instanz (G = (V, E), k) erzeuge den Graphen G′ = (V, E ′ ) +mit E ′ := { (u, v), (v, u) | {u, v} ∈ E }. Gebe (G′ , k) als F EEDBACK V ERTEX S ET Instanz aus. +Korrektheit: +⇒ (Falls (G, k) eine Ja-Instanz von V ERTEX C OVER ist, dann ist (G′ , k) eine Ja-Instanz von +F EEDBACK V ERTEX S ET.): +Sei C ⊆ V ein Vertex Cover mit Größe kleiner oder gleich k. Dann enthält G \C keine Kanten. +Damit hat G′ \C ebenfalls keine Kanten und somit keine Kreise. +⇐ (Falls (G′ , k) eine Ja-Instanz von F EEDBACK V ERTEX S ET ist, dann ist (G, k) eine Ja-Instanz +von V ERTEX C OVER.): +Sei X ⊆ V ′ ein Feedback Vertex Set mit Größe kleiner oder gleich k. Dann ist X ein Vertex +Cover in G, da ansonsten eine Kante {u, v} ∈ E existiert, für die u ∈ + / X ∧v ∈ + / X gilt. Damit wäre +aber (u, v, u) ein Kreis in G′ und X somit kein Feedback Vertex Set. +Laufzeit: Das Erzeugen der Menge E ′ geht in Zeit O(|E|). Insgesamt ist die Reduktion dem- +nach in Laufzeit O(|E|), also in polynomieller Laufzeit, durchführbar. +Also existiert eine polynomielle Reduktion von V ERTEX C OVER auf F EEDBACK V ERTEX S ET. +Da V ERTEX C OVER nach den aktuellen Präsenzaufgaben NP-vollständig ist und +F EEDBACK V ERTEX S ET ∈ NP gilt, ist somit F EEDBACK V ERTEX S ET NP-vollständig. + Bewertung 2 Punkte für die Konstruktion, 1 Punkt für den 1. Fall, 1 Punkt für den 2. Fall, +0.5 Punkte für die Laufzeit, 0.5 Punkte für die Beweisführung im Ganzen + +Hausaufgabe 10.2 (Turingmaschinen) (5 Punkte) +Entwerfen Sie eine Turingmaschine für die Sprache L = { w ∈ Σ∗ | w ist ein Palindrom } über +einem gegebenen Alphabet Σ und geben Sie die Laufzeit Ihrer Turingmaschine an. Begründen +Sie dabei, warum Ihre Turingmaschine korrekt ist und die angegebene Laufzeit hat. + +Lösung. Die Turingmaschine sieht folgendermaßen aus: + (1) Falls kein Wort auf dem Band steht, akzeptiere. + (2) Falls nur ein Buchstabe auf dem Band steht, akzeptiere. + (3) Lese den ersten Buchstaben und bewege danach den Kopf zum letzten Buchstaben. Falls + die beiden Buchstaben verschieden sind, verwerfe. Ansonsten ersetze beide Buchstaben + durch ein neues Symbol x. Ersetze dafür zuerst den letzten Buchstaben und bewege dann + den Kopf zum ersten Buchstaben. + (4) Gehe zu Schritt (1). +Korrektheit: Die Turingmaschine vergleicht immer den ersten und den letzten Buchstaben. +Wenn die Buchstaben nicht gleich sind, ist das Wort kein Palindrom und es wird verworfen. +Ansonsten werden die Buchstaben rausgestrichen (siehe Schritt (3)). Diese Prozedur wird so- +lange wiederholt, bis man in der Mitte des Wortes angekommen ist und entweder nur noch ein +Buchstabe oder kein Buchstabe vorhanden ist. In beiden Fällen ist das Wort ein Palindrom und +es wird akzeptiert (siehe Schritte (1) und (2)). +Laufzeit: Die Schritte (1) bis (3) benötigen jeweils eine Laufzeit von O(n), wobei n hier die +Länge der Eingabe bezeichnet. Da in jedem Durchlauf der Schritte (1) bis (3) zwei Buchstaben +raus gestrichen werden oder das Wort akzeptiert oder verworfen wird, gibt es maximal ⌊n/2⌋ +Durchläufe, bis nur noch ein oder kein Buchstabe auf dem Band stehen und somit in den Schrit- +ten (1) oder (2) akzeptiert wird. Damit ergibt sich insgesamt eine Laufzeit von O(n2 ). + + +Bewertung 2 Punkte für die Turingmaschine, 1 Punkt für die Korrektheit und 2 Punkte für +die Laufzeit: 1 Punkt Angabe, 1 Punkt Begründung + + + + +Abgabe: 23.06.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 10 + Marek Lenczewski + Matrikelnummer: 1025252 + + 22. Juni 2025 + + +Hausaufgabe 10.1 +Teil 1: FeedbackVertexSet ∈ NP + Zuerst brauchen wir einen polynomiellen Verifizierer, um zu zeigen dass FeedbackVertexSet +∈ NP. + + Der Verifizierer bekommt einen Graphen G und eine Teilmenge X, dieser arbeitet wie folgt: + + • Prüfe, ob |X| ≤ k, sonst ablehnen + + • Den Graphen G′ = G \ X konstruieren + + • Tiefensuche auf G′ durchführen, dabei werden Knoten vom aktuellen Pfad als grau mar- + kiert und nach vollständiger Bearbeitung als schwarz. Wird eine Kante zu einem grauen + Knoten gefunden, zeigt dies einen Kreis an und wir lehnen ab + + • Akzeptieren, wenn die Tiefensuche nichts findet + + Die Schritte 1-4 laufen in polynomieller Zeit, somit läuft der Verifizierer in polynomieller +Zeit. Damit ist F eedbackV ertexSet ∈ N P . + + Teil 2: FeedbackVertexSet ist NP-schwer + Jetzt brauchen wir eine Reduktion von VertexCover auf FeedbackVertexSet, um zu zeigen, +dass F eedbackV ertexSet N P − Schwer ist. + + Konstruktion: Sei dafür ein VertexCover gegeben mit einem Graphen G und einer Zahl k. +Aus G wird ein G′ , wobei jede ungerichtete Kante {u, v} ∈ E durch zwei gerichteten Kanten +(u, v) und (v, u) zu E ′ ersetzt wird. Das k wird einfach übernommen k = k ′ . + + Beweis VertexCover nach FeedbackVertexSet: + + • Sei C ein VertexCover von G mit |C| ≤ k + + • Da C ein VertexCover ist, wird jede Kante mit C verbunden + + • Wenn C entfernt wird, dann hat G \ C keine Kanten mehr + + • Dann hat auch G′ \ C keine Kanten und ist somit Kreisfrei + + • Damit ist C ein FeedbackVertexSet für G′ mit |C| ≤ k ′ + + + + Beweis FeedbackVertexSet nach VertexCover: + + 1 + • Sei X ein FeedbackVertexSet von G′ mit |X| ≤ k ′ + + • Dann ist G′ \ X kreisfrei + + • Angenommen X wäre kein VertexCover von G, dann gäbe es eine Kante {u, v} ∈ E mit + u, v ∈ + /X + + • Somit wären u, v noch in G′ \ X vorhanden + + • Die Kanten (u, v) und (v, u) würden, durch die Konstruktion, einen Kreis erzeugen + + • Das ist ein Widerspruch, somit ist G′ \ X kreisfrei und X ist ein VertexCover + + Da F eedbackV ertexSet ∈ N P und N P − schwer ist, folgt dass F eedbackV ertexSet NP- +Vollständig ist. □ + + +Hausaufgabe 10.2 +Lösung. Die Turingmaschine arbeitet wie folgt: + + 1. Falls das Band leer ist oder nur aus Markierungen besteht, akzeptiere + + 2. Merke das erste unmarkierte Zeichen und ersetze es durch X + + 3. Laufe zum letzten unmarkierten Zeichen oder verwefe + + 4. Falls dieses Zeichen nicht mit dem gemerkten übereinstimmt, verwerfe + + 5. Ersetze es durch X und gehe zurück zum Anfang + + 6. Gehe zu 1 + + Korrektheit: Ein Wort ist genau dann ein Palindrom, wenn es sich von vorne und hin- +ten gleich liest. Dies prüft die Turingmaschine, indem sie Schrittweise die äußeren Buchstarben +verifiziert, bis nur noch einer oder keiner Vorhanden ist, dann akzeptiert ist und wenn eine Un- +stimmigkeit auftritt, dann lehnt sie ab. + + Laufzeit: Die Turingmaschine hat eine Laufzeit von O(n2 ). In jedem Durchlauf macht sie +O(n) Schritte und das mit O(n) Durchläufen. □ + + + + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 23.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 11 + + +Definition 1 (Dreiecksüberdeckung eines Graphen (∆ C OVER)). Gegeben ist ein endlicher un- +gerichteter Graph G = (V, E) und eine Zahl k ∈ N. Dieser Graph hat eine Dreiecksüberdeckung, +wenn eine Menge von Knoten C∆ ⊆ V mit |C∆ | ≤ k existiert, so dass zu jedem Teilgraphen (Drei- +eck) D = (VD , ED ) ⊆ (V, E) mit |VD | = 3 und ED = { {v1 , v2 } ∈ VD ×VD | v1 ̸= v2 } von G gilt, +v ∈ C∆ für mindestens ein v ∈ VD . + +Hausaufgabe 11.1 (10 Punkte) + + + (a) (1 Punkt) Markieren Sie im folgenden Graphen (Abbildung 1 (a)) ein V ERTEX C OVER + der Größe 5. + (b) (1 Punkt) Markieren Sie im folgenden Graphen (Abbildung 1 (b)) eine Dreiecksüberde- + ckung der Größe 5. + (c) (8 Punkte) Zeigen Sie, dass ∆ C OVER NP-vollständig ist. + + + + + (a) (b) + + Abbildung 1: Beispielgraph + + +Lösung. + (a) (b) + + Abbildung 2: Beispielgraph Lösung + + +(c) Wir zeigen zunächst, ∆ C OVER ∈ NP mittels eines polynomiellen Verifizierers. + Der polynomielle Verifizierer erhält als Zertifikat eine Menge von Knoten C∆ ⊆ V . Zunächst + überprüft er, ob |C∆ | ≤ k. Falls dies nicht der Fall ist, verwerfe die Instanz. Ansonsten überprüfe + für alle Teilgraphen D = (VD , ED ) ⊆ (V, E) mit |VD | = 3 und ED = { {v1 , v2 } ∈ VD ×VD | v1 ̸= v2 } + von G, ob für mindestens ein v ∈ VD auch v ∈ C∆ gilt. Falls dies gilt, akzeptiere. Ansonsten ver- + werfe. + Korrektheit: Der Verifizierer überprüft die geforderten Eigenschaften. Falls ein Graph eine Drei- + ecksüberdeckung besitzt, so existiert ein entsprechendes Zertifikat C∆ und der Verifizierer ak- + zeptiert die Instanz. Falls ein Graph keine Dreiecksüberdeckung besitzt, exisitert kein valides + entsprechendes Zertifikat C∆ und der Verifizierer verwirft die Instanz. + Laufzeit: Es gibt maximal |V |3 verschiedene Teilmengen, die ein Dreieck bilden könnten und + daher überprüft werden müssen. Die Überprüfung geht in O(|V |) Zeit. Da alle anderen Schritte + des Verifizierers in O(1) Zeit erledigt werden können, ergibt sich somit insgesamt eine Laufzeit + von O(|V |4 ), was einer polynomiellen Laufzeit entspricht. + Also gilt ∆ C OVER ∈ NP. + Nun reduzieren wir V ERTEX C OVER auf ∆ C OVER. Betrachte folgende Reduktion: + Sei I = (G = (V, E), k) eine Instanz von V ERTEX C OVER. Bilde den Graphen G′ = (V ′ , E ′ ) mit + V ′ = V ∪ { ve | e ∈ E } und E ′ = E ∪ { {ve , e1 }, {ve , e2 } | e = {e1 , e2 } ∈ E }. Gebe I ′ = (G′ , k) + als ∆ C OVER Instanz aus. + Korrektheit: + ⇒ (Falls (G, k) eine Ja-Instanz von V ERTEX C OVER ist, dann ist (G′ , k) eine Ja-Instanz von ∆ + C OVER.): + Sei C ein Vertex Cover mit Größe kleiner oder gleich k von G. Dann ist C ebenfalls eine Drei- + ecksüberdeckung mit Größe kleiner oder gleich k. Offensichtlich gilt |C| ≤ k. Da C ein Vertex + Cover von G ist, gilt für jede Kante {u, v} ∈ E, dass u ∈ C oder v ∈ C. Für alle Teilgraphen + D = (VD , ED ) ⊆ (V, E) mit |VD | = 3 und ED = { {v1 , v2 } ∈ VD ×VD | v1 ̸= v2 } von G gilt damit + direkt, dass mindestens ein v ∈ VD auch in C enthalten ist. Der Graph G′ besitzt neben allen + solchen Teilgraphen in G zusätzlich solche Teilgraphen (Dreiecke), die durch die Knotenmenge + { ve | e ∈ E } und die entsprechenden Kanten { e, {ve , e1 }, {ve , e2 } | e = {e1 , e2 } ∈ E } entste- +hen. Für diese Dreiecke gilt aber auch, dass mindestens ein Knoten, nämlich e1 oder e2 , in C +enthalten sein muss, da C ein Vertex Cover ist. +⇐ (Falls (G′ , k) eine Ja-Instanz von ∆ C OVER ist, dann ist (G, k) eine Ja-Instanz von V ERTEX - +C OVER.): +Sei C∆ eine Dreiecksüberdeckung von G′ der Größe |C∆ | ≤ k. Ohne Beschränkung der All- +gemeinheit können wir davon ausgehen, dass C∆ ⊆ V gilt: Falls für einen Knoten ve ∈ C∆ gilt, +können wir e1 zu C∆ hinzufügen und ve aus C∆ löschen. Dadurch bleibt die Überdeckungseigen- +schaft erhalten und die Menge wird höchstens kleiner. Nun gilt also C∆ ⊆ V . Die Menge C∆ ist +auch ein Vertex Cover mit Größe kleiner oder gleich k von G. Offensichtlich gilt |C∆ | ≤ k. Da für +alle Teilgraphen D′ = (VD′ , ED′ ) ⊆ (V ′ , E ′ ) mit |VD′ | = 3 und ED′ = { {v1 , v2 } ∈ VD′ ×VD′ | v1 ̸= v2 } +von G′ gilt, dass mindestens ein Knoten v ∈ VD′ ebenfalls in C∆ enthalten sein muss, gilt dies +auch für die Dreiecke {ve , e1 , e2 }. Da außerdem C∆ ⊆ V gilt, gilt e1 ∈ C∆ oder e2 ∈ C∆ für alle +Kanten e ∈ E. +Laufzeit: Das Hinzufügen von einem neuen Knoten und zwei neuen Kanten pro Kante geht in +O(|E|). Damit geht die gesamte Reduktion in O(|E|), also in polynomieller Zeit. +Also existiert eine polynomielle Reduktion von V ERTEX C OVER auf ∆ C OVER. Da V ERTEX - +C OVER NP-vollständig aus den Präsenzaufgaben bekannt ist und ∆ C OVER ∈ NP gilt, ist somit +∆ C OVER NP-vollständig. + + +Bewertung (a) und (b) jeweils 1 Punkt, (c) gibt 8 Punkte: 1 Punkt Angabe Verifizierer/NDTM, +0,5 Punkte Korrektheit Verifizierer/NDTM, 0,5 Punkte Laufzeit Verifizierer/NDTM, 1 Punkt +Angabe Reduktion, 3 Punkte Korrektheit (je Richtung 1,5 Punkte), 1 Punkt Laufzeit Reduktion, +1 Punkt Beweisführung + + + + +Abgabe: 30.06.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 11 + Marek Lenczewski + Matrikelnummer: 1025252 + + 28. Juni 2025 + + +Hausaufgabe 11.1 +a + 1 + + + 2 3 + + + 4 5 + + + 6 10 + + + 7 9 + + + 8 + + +b + a b c + + + d e + f g + + h i j + + + k l m n + o p q + r s + + + t u v + + +c +Teil 1: ∆-Cover ∈ NP + + + 1 + Zuerst brauchen wir einen polynomiellen Verifizierer, um zu zeigen, dass ∆-Cover ∈ NP. +Der Verifizierer bekommt einen Graphen G = (V, E), Zahl k und ein Zertifikat C∆ ⊆ V und +arbeitet wie folgt: + + + 1. Prüfe, ob |C∆ | ≤ k + 2. Prüfe alle Knotentripel {a, b, c} ⊆ V , ob sie ein Dreieck bilden (also ob {a, b}, {b, c} und + {a, c} ∈ E zutrifft) und ob a, b oder c ∈ C∆ + 3. Ablehnen, falls ein Dreieck den test nicht besteht, sonst akzeptieren + + Die Schritte 1 und 3 laufen in O(1) und Schritt zwei läuft in O(|V |3 ), das ergibt eine poly- +nomielle Laufzeit. Damit ist ∆-Cover ∈ NP. + + Teil 2: ∆-Cover ist NP-schwer + Jetzt brauchen wir eine Reduktion von VertexCover auf ∆-Cover, um zu zeigen, dass +∆-Cover NP-schwer ist. + + Konstruktion: Sei dafür ein VertexCover gegeben mit einem Graphen G = (V, E) und +einer Zahl k. Aus G wird G′ = (V ′ , E ′ ), wobei für jede Kante e = {u, v} ∈ E ein neuer Kno- +ten we ∈ V ′ hinzugefügt wird und mit beiden Knoten u, v verbunden wird, sodass die Kanten +{v, we } und {u, we } zu E ′ dazu kommen. Das k = k ′ wird übernommen. So werden aus allen +Kanten Dreiecke. + + Beweis VertexCover nach ∆-Cover + • Sei C ein VertexCover für G mit |C| ≤ k + • Zu zeigen sei, dass C auch ein ∆-Cover für G′ mit |C| ≤ k ist + • Da C ein VertexCover ist, gilt u ∈ C oder v ∈ C + • Jedes Dreieck in G′ hat die Form {u, v, we } für eine Kante e = {u, v} ∈ E + • Somit hat jedes Dreieck mindestens einen Knoten in C + • Also ist C ein ∆-Cover für G′ + Beweis ∆-Cover nach VertexCover + • Sei C∆ ein ∆-Cover für G′ mit |C∆ | ≤ k ′ + • Jede Kante e = {u, v} ∈ E bildet durch {u, v, we } ein Dreieck in G′ + • Also gilt u, v oder we ∈ C∆ für alle Kanten in G′ + • Falls we ∈ C∆ , dann kann man we durch u oder v tauschen ohne die Eigenschaft von + ∆-Cover zu verletzen, sodass u oder v ∈ C∆ für alle Kanten in G′ gilt + • Da sich die Knoten aus C∆ nur vertauschen gilt weiter |C| = |C∆ | ≤ k + • Weiter sind in C∆ nur noch die Knoten aus V , somit gilt u ∈ C oder v ∈ C für alle Kanten + E + • Also ist C ein VertexCover für G + + + + Da ∆-Cover ∈ NP und NP-schwer ist, folgt dass ∆-Cover NP-vollständig ist. □ + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 30.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 12 + + +Hausaufgabe 12.1 (Lower Bound: V ERTEX C OVER) (5 Punkte) +Betrachten Sie die Reduktion aus der Musterlösung zu Präsenzserie 10 von C LIQUE auf V ER - +TEX C OVER . + + 1. Geben Sie die Anzahl der Knoten, die Anzahl der Kanten und den Wert k für die re- + sultierende V ERTEX C OVER Instanz in Abhängigkeit von der originalen C LIQUE Instanz + an. + 2. Zeigen Sie Lower Bounds für V ERTEX C OVER bezüglich Anzahl der Knoten, Anzahl + der Kanten und k basierend auf der ETH und der Reduktion aus der Musterlösung zu + Präsenzserie 10. + +Lösung. 1. Sei I = (G = (V, E), k) eine Instanz von C LIQUE und I ′ = (G′ = (V ′ , E ′ ), k′ ) die + aus der Reduktion resultierende Instanz von V ERTEX C OVER. Es gilt |V ′ | = |V |, |E ′ | = + 1 2 ′ + 2 |V |(|V | − 1) − |E| ≤ |V | und k = |V | − k ≤ |V |. + ′ + 2. • Angenommen es gäbe einen Algorithmus, der V ERTEX C OVER in 2o(|V |) · |I|O(1) + löst. Durch die Kombination der Reduktion von C LIQUE auf V ERTEX C OVER und + diesem Algorithmus, existiert dann auch ein 2o(|V |) ·|I|O(1) Algorithmus für C LIQUE, + da |V ′ | = |V | nach Aufgabenteil 1. Aus der Musterlösung der Präsenzserie 12 folgt + somit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für V ERTEX C OVER nur geben, wenn die ETH + falsch ist. + √ ′ + • Angenommen es gäbe einen Algorithmus, der V ERTEX C OVER in 2o( |E |) · |I|O(1) + löst. Durch die Kombination der Reduktion von C LIQUE auf V ERTEX C OVER und + diesem Algorithmus, existiert dann auch ein 2o(|V |) ·|I|O(1) Algorithmus für C LIQUE, + da |E ′ | ≤ |V |2 nach Aufgabenteil 1. Aus der Musterlösung der Präsenzserie 12 folgt + somit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für V ERTEX C OVER nur geben, wenn die ETH + falsch ist. + ′ + • Angenommen es gäbe einen Algorithmus, der V ERTEX C OVER in 2o(k ) · |I|O(1) löst. + Durch die Kombination der Reduktion von C LIQUE auf V ERTEX C OVER und die- + sem Algorithmus, existiert dann auch ein 2o(|V |) · |I|O(1) Algorithmus für C LIQUE, + da k′ ≤ |V | nach Aufgabenteil 1. Aus der Musterlösung der Präsenzserie 12 folgt + somit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für V ERTEX C OVER nur geben, wenn die ETH + falsch ist. + Bewertung 1 Punkt für den ersten Teil und 1.5 + 1.5 + 1 für den zweiten. + +Definition 1 (H ITTING S ET). Bei diesem Problem ist ein Universum U, Teilmengen des Uni- +versums F1 , . . . , Fr ⊆ U und eine Zahl k ∈ N≥0 gegeben. Gibt es eine Menge S ⊆ U mit |U| ≤ k +und S ∩ Fi ̸= 0/ für alle i ∈ [r]? +Anschaulicher wird bei diesem Problem nach maximal k Elementen aus dem Universum ge- +fragt, sodass diese Elemente alle gegebenen Mengen Fi „treffen“. + +Hausaufgabe 12.2 (Lower Bound: H ITTING S ET) (5 Punkte) +Betrachten Sie folgende Reduktion von 3-SAT auf H ITTING S ET. +Für eine gegebene 3-SAT Instanz I mit n Variablen x1 , . . . , xn und m Klauseln C1 , . . . ,Cm : Als +Universum wählen wir U = { x1 , x̄1 , . . . , xn , x̄n }. Für jede Variable xi fügen wir die Menge Fi := +{ xi , x̄i } hinzu. Für jede Klausel C j fügen wir die Menge Fn+ j := C j hinzu. Wir setzen k := n +und erhalten damit eine H ITTING S ET Instanz I. + 1. Geben Sie die Größe des Universums, die Anzahl der Mengen und die Zahl k aus der + resultierenden H ITTING S ET Instanz in Abhängigkeit von der Anzahl der Variablen n und + der Anzahl der Klauseln m aus der originalen 3-SAT Instanz an. + 2. Beweisen Sie Lower Bounds für H ITTING S ET bezüglich Universumsgröße, Anzahl der + Mengen und der Zahl k basierend auf der obigen Reduktion und der ETH. + +Lösung. 1. Für die Universumsgröße gilt |U| = 2n, für die Anzahl der Mengen gilt r = + n + m und es gilt k = n. + 2. • Angenommen es gäbe einen Algorithmus, der H ITTING S ET in 2o(|U|) · |I|O(1) löst. + Durch die Kombination der Reduktion von 3-SAT auf H ITTING S ET und diesem + Algorithmus, existiert dann auch ein 2o(n) · |I|O(1) Algorithmus für 3-SAT, da |U| = + 2n nach Aufgabenteil 1. Damit ist die ETH falsch. + Damit kann es so einen Algorithmus für H ITTING S ET nur geben, wenn die ETH + falsch ist. + • Angenommen es gäbe einen Algorithmus, der H ITTING S ET in 2o(r) · |I|O(1) löst. + Durch die Kombination der Reduktion von 3-SAT auf H ITTING S ET und diesem + Algorithmus, existiert dann auch ein 2o(m) · |I|O(1) Algorithmus für 3-SAT, da r = + n + m nach Aufgabenteil 1 und n ≤ 3m für 3-SAT gilt. Aus dem Sparsification + Lemma folgt damit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für H ITTING S ET nur geben, wenn die ETH + falsch ist. + • Angenommen es gäbe einen Algorithmus, der H ITTING S ET in 2o(k) · |I|O(1) löst. + Durch die Kombination der Reduktion von 3-SAT auf H ITTING S ET und diesem + Algorithmus, existiert dann auch ein 2o(n) · |I|O(1) Algorithmus für 3-SAT, da k = n + nach Aufgabenteil 1. Damit ist die ETH falsch. + Damit kann es so einen Algorithmus für H ITTING S ET nur geben, wenn die ETH + falsch ist. + + +Bewertung 1 Punkt für den ersten Teil und 1.5 + 1.5 + 1 für den zweiten. + +Abgabe: 07.07.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 12 + Marek Lenczewski + Matrikelnummer: 1025252 + + 6. Juli 2025 + + +Hausaufgabe 12.1.1 +Die Knotenzahl bleibt gleich: |V ′ | = |V | = n +Die Kantenzahl entspricht den nicht vorhandenen Kanten: |E ′ | = |E| +k ′ entspricht allen Knoten, die vorher nicht in der Clique waren: k ′ = n − k + + +Hausaufgabe 12.1.2 +Wir wissen, dass 3-SAT laut ETH mindestens 2Ω(n) braucht, wobei n für die Anzahl der Varia- +blen steht. + + In der Vorlesung wurde eine Reduktion von 3-SAT auf Clique gezeigt. In der Präsenzaufgabe +10.2 wurde eine Reduktion von Clique auf VertexCover gezeigt. Also gibt es eine Reduktion von +3-SAT zu VertexCover. Somit kann VertexCover nicht schneller lösbar sein als 3-SAT, weil sonst +3-SAT schneller werden würde, was nach ETH nicht geht. + + So ergeben sich die folgenden Lower Bounds: + + • Bezüglich Anzahl der Knoten: 2Ω(n) , wobei n = |V ′ |. Die Knotenzahl verändert sich + nicht bei den Reduktionen. + √ + • Bezüglich Anzahl der Kanten: 2Ω( m) , wobei m = |E ′ |. Die Kanten können bei den + Reduktionen zu einem quadratischen Wachstum der Kanten führen. + ′ + • Bezüglich k ′ : 2Ω(k ) . Bei der Reduktion gilt k ′ = n − k, wobei k ′ linear von n abhängt. + + + + + 1 + Hausaufgabe 12.2.1 + • Größe des Universums: |U | = 2n. Für jede der n Variablen gibt es zwei Elemente (xi + und xi ) + + • Anzahl der Mengen: n + m. n sind die Variablenmengen und m die Klauselmengen. (Fi + und Fn+j ) + + • Parameter: k = n. Wird in der Reduktion so übernommen. + + +Hausaufgabe 12.2.2 +Wir wissen, dass 3-SAT laut ETH mindestens 2Ω(n) braucht, wobei n für die Anzahl der Varia- +blen steht. Somit kann HittingSet nicht schneller sein. + + Es ergeben sich die folgenden Lower Bounds für HittingSet: + + • Bezüglich Größe des Universums: 2Ω(|U |/2) , da für jede Variable zwei Elemente ins + Universum aufgenommen werden + + • Bezüglich Anzahl der Mengen: 2Ω(n) , wobei die Anzahl der Variablen- und Klausel- + mengen (m + n) mindestens n ist + + • Bezüglich Zahl k: 2Ω(k) , da in der Reduktion k = n gilt. + + + + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 07.07.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 13 + + +Definition 1 (M AX -3-S AT). Bei dem Problem M AX -3-S AT ist eine Formel φ in konjunktiver +Normalform gegeben, wobei jede Klausel 3 Literale enthält. Gesucht ist eine Belegung β der +Variablen, die die Anzahl v(β ) der erfüllten Klauseln maximiert. + +Hausaufgabe 13.1 (M AX -3-S AT) (10 Punkte) +Betrachten Sie folgenden Algorithmus A: + • Sei β0 die Belegung, die alle Variablen auf false setzt. + • Sei β1 die Belegung, die alle Variablen auf true setzt. + • Falls v(β0 ) ≥ v(β1 ), gib β0 zurück. + • Sonst gib β1 zurück. + + (a) (6 Punkte) Zeigen Sie, dass obiger Algorithmus Güte 2 hat, d.h. v(A(φ )) ≥ 12 v(OPT (φ )) + gilt für alle Eingaben φ . + (b) (4 Punkte) Geben Sie eine Formel an, bei der der Algorithmus eine Belegung ausgibt, die + genau die Hälfte der maximal erfüllbaren Klauseln erfüllt. Begründen Sie kurz, warum + Ihre Formel geeignet ist. + Hinweis: Sie brauchen nur höchstens zwei Klauseln und vier Variablen. + +Lösung. (a) Für jede Klausel C gilt, dass sie von mindestens einer der Belegungen β0 und + β1 erfüllt wird. Wir zeigen nun per Widerspruch, dass der obige Algorithmus eine Güte + von 2 hat: + Angenommen, es gilt v(A(φ )) < 12 v(OPT (φ )) für eine Formel φ . Sei m die Anzahl an + Klauseln in φ . Dann gilt v(OPT (φ )) ≤ m. Da v(A(φ )) < 12 v(OPT (φ )) ≤ 12 m, gilt v(β0 ) < + 1 1 + 2 m und v(β1 ) < 2 m. Da jede Klausel C allerdings von mindestens einer der Belegungen + β0 und β1 erfüllt wird, gilt v(β0 ) ≥ 12 m oder v(β1 ) ≥ 12 m. Dies ist ein Widerspruch. + Also gilt v(A(φ )) ≥ 21 v(OPT (φ )) für alle Eingaben φ und obiger Algorithmus hat Güte + 2. + (b) Betrachte folgende Formel: φ = (x1 ∨x2 ∨x3 )∧(¬x0 ∨¬x2 ∨¬x3 ) Mit β = {x0 → f alse, x1 → + true, x2 → true, x3 → true} gilt v(β ) = 2. Da φ nur zwei Klauseln hat, gilt somit v(OPT (φ )) = + 2. Es gilt allerdings v(β0 ) = 1, da hier nur die zweite Klausel erfüllt ist, und v(β1 ) = 1, + da hier nur die erste Klausel erfüllt ist. Somit gilt v(A(φ )) = 1 = 12 · 2 = 21 v(OPT (φ )). + Bewertung (a) 2 Punkte korrekte Beweisführung im Allgemeinen, 2 Punkte Abschätzung +Algorithmus, 2 Punkte Abschätzung Optimum; (b) 2 Punkte für Angabe einer korrekten Formel, +1 Punkt Ausgabe Algorithmus, 1 Punkt Optimum + + + + +Abgabe: 14.07.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 13 + Marek Lenczewski + Matrikelnummer: 1025252 + + 13. Juli 2025 + + +Hausaufgabe 13.1.a +Zu zeigen ist, dass der Algorithmus die Güte 2 hat, also die Lösung vom Algorithmus mindestens +halb so gut ist wie die optimale Lösung. Also ν(A(ϕ)) ≥ 21 ν(OP T (ϕ)) für alle Eingaben ϕ. + + Sei ϕ eine beliebige 3-SAT-Formel mit m Klauseln C1 , C2 , . . . , Cm , wobei jede Klausel genau +3 Literale enthält. + + Jede Klausel Ci ist wahr, wenn β0 wahr ist oder wenn β1 wahr ist oder beides. Da die Varia- +blen in Ci nur positiv oder negativ sein können, muss dabei mindestens β0 oder β1 wahr werden. + + Der Algorithmus wählt die Belegung β0 oder β1 aus, die die meisten Klauseln Ci erfüllt. +Somit ist die Anzahl der ausgewählten Klauseln Ci mindestens m2 , da eine bei jeder Klausel +mindestens eine Belegung wahr werden muss und somit hat mindestens eine Belegung mindes- +tens m + 2 wahre Klauseln. + + Sei ν0 = Anzahl der von β0 erfüllten Klauseln und ν1 = Anzahl der von β1 erfüllten Klauseln. + + So ergibt sich ν(A(ϕ)) = max{ν0 , ν1 } ≥ ν0 +ν + 2 + 1 + ≥m + 2. + + Da die optimale Lösung höchstens alle m Klauseln erfüllen kann, gilt ν(OP T (ϕ)) ≤ m. + ν(OP T (ϕ)) + Daraus folgt: ν(A(ϕ)) ≥ m + 2 ≥ 2 . + + Somit hat der Algorithmus eine Güte von 2. + + + + + 1 + Hausaufgabe 13.1.b +Der Algorithmus erfüllt genau die Hälfte der maximal erfüllbaren Klauseln, wenn jede Klausel +nur aus positiven oder negativen Variablen besteht und sich diese genau gleich aufteilen. + + Formel: ϕ = (x1 ∨ x2 ∨ x3 ) ∧ (¬x1 ∨ ¬x2 ∨ ¬x3 ) + + Unter β0 (alle false): + + • C1 = (x1 ∨ x2 ∨ x3 ) = (false ∨ false ∨ false) = false + + • C2 = (¬x1 ∨ ¬x2 ∨ ¬x3 ) = (true ∨ true ∨ true) = true + + • ν(β0 ) = 1 + + Unter β1 (alle true): + + • C1 = (x1 ∨ x2 ∨ x3 ) = (true ∨ true ∨ true) = true + + • C2 = (¬x1 ∨ ¬x2 ∨ ¬x3 ) = (false ∨ false ∨ false) = false + + • ν(β1 ) = 1 + + Der Algorithmus wählt ν(A(ϕ)) = max{ν(β0 ), ν(β1 )} = max{1, 1} = 1. + + Eine optimale Lösung wäre zum Beispiel x1 = true, x2 = false, x3 = false. + + • C1 = (true ∨ false ∨ false) = true + + • C2 = (false ∨ true ∨ true) = true + + Daraus ergibt sich ν(OP T (ϕ)) = 2, also ν(A(ϕ)) = ν(OP2T (ϕ)) . 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Jansen + + 19. Juli 2024 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2024 + Name: Matrikel-Nr.: + + + Hinweise: + + • Bearbeiten Sie alle Aufgaben. + • Sie haben 10 Minuten Einlesezeit. Danach haben Sie 120 Minuten Zeit für die Klausurbearbeitung. + • Erlaubte Hilfsmittel: Ein doppelseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein + Rot, kein Grün, kein Bleistift). + • Einsichtnahme: 18.08.2024. + Note 4,0 3,7 3,3 3,0 2,7 2,3 2,0 1,7 1,3 1,0 + • + Mindestpunkzahl 17,5 19,5 21,5 23,5 25 27 29 31 33 35 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 12 + Aufgabe 1 KOMPLEXITÄT: A NWENDUNG LPT (4+2+2+2 Punkte) + Gegeben ist folgende Scheduling Instanz: + Jj J1 J2 J3 J4 J5 J6 J7 + m=3 + pj 5 3 7 8 1 4 2 + + a) Wenden Sie den LPT-Scheduling Algorithmus auf die obige Instanz an. Geben Sie dafür alle + relevanten Zwischenschritte an, sowie die Reihenfolge in der die Jobs auf Maschinen platziert + werden. Stellen Sie den resultierende Schedule graphisch dar. Eine beispielhafte Darstellung + ist: + + J3 + J6 + J2 + J5 J7 + J1 + J4 + m1 m2 m3 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 12 + b) Geben Sie die approximative Güte des Algorithmus LPT an. + + + + + c) Geben Sie eine Instanz für m = 2 Maschinen an, bei denen der Algorithmus ListScheduling + eine Güte von 2 − 1/m erzielt. Geben Sie zusätzlich an, welche Werte die optimale Lösung + OPT und die Lösung von LPT für diese Instanz erzielen. + + + + + d) Geben Sie eine Instanz für m = 3 Maschinen an, bei denen der Algorithmus ListScheduling + eine Güte von 2 − 1/m erzielt. Geben Sie zusätzlich an, welche Werte die optimale Lösung + OPT und die Lösung von LPT für diese Instanz erzielen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 12 + Aufgabe 2 KOMPLEXITÄT: B EWEIS : K -C LIQUE (10 Punkte) + Beweisen Sie folgende Aussage aus der Vorlesung: + SAT ≤ k-Clique. + Hinweis: NICHT 3-SAT, sondern allgemeines SAT! + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 12 + Aufgabe 3 KOMPLEXITÄT: NP-VOLLSTÄNDIGKEIT (5+5 Punkte) a) + Wir betrachten folgendes Problem: + Problem: CliqueAndIndependetSet + Eingabe: Ein ungerichteter Graph G = (V, E) mit V = 1, . . . , |V | und eine ganze Zahl k ≥ 1. + Entscheide: Gibt es in G sowohl ein Independent Set (Menge von Knoten, die paarweise + nicht adjazent sind, d.h. zwischen denen es keine Kanten gibt) der Größe k und als auch eine + Clique der Größe k? + Zeigen Sie die NP-Schwere von CliqueAndIndependetSet durch Angabe einer Reduktion + eines nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 12 + b) Wir betrachten folgendes Problem: + Problem: (a1 = 1)-SubsetSum + Eingabe: Eine Menge von n Items, jedes Item i ∈ [n] hat eine Größe ai ∈ N>0 , wobei a1 = 1, + und ein Zielwert T . + Entscheide: Gibt es I ⊆ [n] sodass ∑i∈I ai = T ? + Zeigen Sie die NP-Schwere von (a1 = 1)-SubsetSum durch Angabe einer Reduktion eines + nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 12 + Aufgabe 4 KOMPLEXITÄT: A PPROXIMATIVE A LGORITHMEN (3+7 Punkte) + Wir betrachten folgendes Problem: + Problem: A PPROXIMATE S UBSET S UM + Eingabe: n ganze Zahlen a1 , . . . an und ein Zielwert T . Es gilt ai ≤ T ∀i ∈ [n] sowie ∑i∈[n] ai > + T. + Ziel: Wähle Zahlenmenge S ⊆ {1, . . . n} mit ∑ j∈S a j ≤ T und maximiere ∑ j∈S a j . + + Betrachten Sie zu diesem Problem den folgenden Algorithmus GA. + + Algorithmus GA((A,T)) + + 1 sortiere Zahlen absteigend, sodass a1 ≥ a2 ≥ · · · ≥ an gilt; + 2 integer i = 1; + 3 integer sum = 0; + 4 boolean tooLarge = false; + 5 while i ≤ n and tooLarge = false do + 6 if sum+ai ≤ T then + 7 sum=sum+ai ; + 8 i = i + 1; + 9 else + 10 tooLarge = true; + 11 fi + 12 od + 13 return sum; + + a) Geben Sie eine Konstruktionsvorschrift für eine Instanz I an, mit der der Algorithmus GA + beliebig nah an eine Güte von 2 kommt, also A(I) ≈ OPT2 (I) . + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 12 + b) Zeigen Sie, dass der Algorithmus GA eine approximative Güte von 2 hat, also A(I) ≥ OPT2 (I) + gilt. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 12 + Aufgabe 5 KOMPLEXITÄT: ETH (5+5 Punkte) + + a) Betrachten Sie nachfolgende Reduktion von 3-SAT auf k-Color: + 1. Eliminiere doppelte Klauseln und Variablen, die in keiner Klausel auftauchen + 2. Elimiere Klauseln die eine Variable und ihre Negation enthalten (diese sind immer erfüllt) + 3. Seien C1 , . . . ,Cm die Klauseln in der SAT Formel + 4. Definiere  + V := { xi , x̄i , vi | i ∈ [n] } ∪ C j j ∈ [m] ∪ {z} , + sowie +  + E := {vi , v j }, {vi , x j }, {vi , x̄ j } i ∈ [n], j ∈ [n] \ {i} + ∪ { {xi , x̄i } | i ∈ [n] } +  + ∪ {xi ,C j } i ∈ [n], j ∈ [m], xi ∈ / Cj +  + ∪ {x̄i ,C j } i ∈ [n], j ∈ [m], x̄i ∈ / Cj + ∪ { {vi , z} | i ∈ [n] } +  + ∪ {C j , z} j ∈ [m] + Welche Lower Bounds ergeben sich unter Annahme der ETH for k-Colour aus dieser Re- + duktion in Hinblick auf + (i) die Anzahl der Knoten |V |; sowie + (ii) die Anzahl der Kanten |E|? + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 12 + b) Wir betrachten folgendes Scheduling Problem: + Problem: 2|prec, pi ∈ {1, 2}|Cmax + Eingabe: Eine Menge J von n Jobs, jeder Job J ∈ J hat eine Ausführungszeit pJ ∈ {1, 2} + sowie Präzedenzconstraints in Form eines gerichteten, azyklischem Graphen G = (J , A), + sowie ein ganze Zahl T ≥ 0. + Entscheide: Existiert ein Schedule, d.h. eine Abbildung, die jedem Job eine Startzeit und + eine Maschine zuweist, sodass (a) sich Jobs auf einer Maschine nicht überlappen (b) für alle + Jobs j, k ∈ J mit ( j, k) ∈ A, dass k erst startet nachdem j vollständig abgearbeitet ist und + (c) der Makespan höchstens T ist? + Wir betrachten die folgende Reduktion von k-Clique auf 2|prec, pi ∈ {1, 2}|Cmax . + Seien G = (V, E) und k gegeben. Wir setzen n = 4|V | + 3|E| und erstellen einen Knoten-Job Ji + für alle i ∈ V mit pi = 1. Des Weiteren erstellen wir Kanten-Jobs J{i, j} für jede Kante {i, j} ∈ E + mit p{i, j} = 2. Abschließend erstellen wir noch 3|V | + 2|E| Dummy Jobs. Diese Jobs haben + Ausführungszeit pdummy = 1. + Wir setzen unsere Präzedenzen nach dem folgenden Schema: + 1) (Ji , J{i, j} ) für alle i ∈ V, {i, j} ∈ E + 2) Präzedenzen innerhalb der Dummy Jobs. + Zuletzt setzen wir den Makespan T = 2|V | + 2|E|. + (i) Geben Sie die Lower Bounds für 2|prec, pi ∈ {1, 2}|Cmax in Hinblick auf n an, die sich + unter der ETH aus dieser Reduktion, basierend auf den Lower Bounds für k-Clique, + ergeben. + (ii) Geben Sie die Lower Bounds für 2|prec, pi ∈ {1, 2}|Cmax in Hinblick auf T an, die sich + unter der ETH aus dieser Reduktion, basierend auf den Lower Bounds für k-Clique, + ergeben. + Hinweis: Ein beispielhafter Schedule ist unten dargestellt. Die Präzedenzen sind im Graphen + durch gerichtete Kanten abgebildet, daher startet J4 erst nach dem Ende von J1 . Nutzen Sie + für diese Aufgabe die bekannten Lower Bounds aus der Präsenzübung. Diese wurden aus der + Reduktion von 3-SAT auf k-Clique hergeleitet. Hierbei ist der Clique Graph zusammenhän- + gend! + + J3 + + J1 + J3 J4 + J4 + + J1 J2 + J2 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 12 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 12 + Aufgabe 6 B ONUSAUFGABE (10 Punkte) + Beweisen oder widerlegen Sie folgende Aussage: + Wenn die ETH fehlschlägt gilt P = NP. + + + + + Viel Erfolg! + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 12 von 12 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen + + 11. Oktober 2024 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2024 + Name: Matrikel-Nr.: + Hinweise: + + • Bearbeiten Sie alle Aufgaben. + • Sie haben 10 Minuten Einlesezeit. Danach haben Sie 120 Minuten Zeit für die Klausurbearbeitung. + • Erlaubte Hilfsmittel: Ein doppelseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein + Rot, kein Grün, kein Bleistift). + • Einsichtnahme: 30.10.2024 + Note 4,0 3,7 3,3 3,0 2,7 2,3 2,0 1,7 1,3 1,0 + • + Mindestpunktzahl 17,5 19,5 21,5 23,5 25 27 29 31 33 35 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 16 + Aufgabe 1 ∆T SP1 (6+2+2 Punkte) + Gegeben sei folgender Graph G = (V, E), bei dem die Kantengewichte jeweils an der nach außen + gerichteten Seite der Kante stehen: + + a + 4 2 + + 3 + e b + 4 2 + + 4 3 + 6 5 + + d c + 2 + + a) Wenden Sie den Algorithmus ∆T SP1 auf den Graphen G beginnend bei a an. Geben Sie dabei + die Ergebnisse jedes Zwischenschrittes des Algorithmus an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 16 + b) Beschreiben Sie, was der Algorithmus ∆T SP1 berechnet und geben Sie die Bedingungen an, + die an den Graphen gestellt sind, damit der Algorithmus eine korrekte Lösung berechnet. + + + + + c) Begründen Sie, warum der Algorithmus ∆T SP1 eine approximative Güte von 2 besitzt. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 16 + Aufgabe 2 KOMPLEXITÄT: VORLESUNGSBEWEIS SAT (10 Punkte) + Beweisen Sie folgende Aussage: + 3-SAT ist NP-vollständig. Zeigen Sie dafür SAT ≤ 3-SAT, nehmen Sie also 3-SAT ∈ NP als + gegeben an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 16 + Aufgabe 3 KOMPLEXITÄT: NP-VOLLSTÄNDIGKEIT (5+5 Punkte) + + a) Wir betrachten folgendes Problem: + Problem: Hitchhiker’s-HamiltonianCycle + Eingabe: Ein ungerichteter Graph G = (V, E) mit V = {1, . . . , |V |} und für alle v ∈ V gilt + deg(v) ≥ 42. + Entscheide: Gibt es einen Hamiltonkreis in G? + Beweisen Sie die NP-Schwere von Hitchhiker’s-HamiltonianCycle durch Angabe einer + Reduktion eines nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + Hinweis: Als Hilfe, geben Sie graphisch die Reduktion für Knotengrad ≥ 10 an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 16 + b) Wir betrachten folgendes Problem: + Problem: AtMostTwoPerSize-SubsetSum + Eingabe: Ein ganzzahliger Zielwert T > 0, eine Menge von n Items, jedes Item i ∈ [n] + hat eine Größe ai ∈ N>0 , jede Größe tritt höchstens zwei Mal auf, d.h. für alle i ∈ [n] gilt + | { i′ ∈ [n] | ai′ = ai } | ≤ 2. + Entscheide: Gibt es S ⊆ [n] sodass ∑i∈S ai = T ? + Beweisen Sie die NP-Schwere von AtMostTwoPerSize-SubsetSum durch Angabe einer Re- + duktion eines nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 16 + Aufgabe 4 KOMPLEXITÄT: A PPROXIMATIVE A LGORITHMEN (10+(10) Punkte) + Wir betrachten das aus der Vorlesung bekannte Makespan Scheduling Problem P||Cmax . Dazu ist + folgender Algorithmus gegeben: + + Algorithmus ROUND ROBIN S CHEDULING(I=(J,m)) + + 1 sortiere die Jobs in J so, dass p1 ≥ p2 ≥ · · · ≥ pn gilt; + 2 setze B1 = · · · = Bm = 0; + / + 3 integer j = 1; + 4 integer i = 1; + 5 while j <= n do + 6 Platziere Job J j auf Maschine Mi ; + 7 Bi = Bi ∪ {J j }; + 8 j = j + 1; + 9 if i < m then + 10 i=i+1 + 11 else + 12 i=1 + 13 fi + 14 od + 15 return B1 , . . . , Bm + + Betrachten Sie folgendes Beispiel zu der Funktionsweise des Round Robin Scheduling Algorith- + mus’. Gegeben sind 3 Maschinen und 6 Jobs J1 , . . . J6 , die bereits richtig sortiert sind. Es gilt + p1 = 7, p2 = 5, p3 = 5, p4 = 4, p5 = 3, p6 = 1. Daraus ergibt sich folgender Schedule: + + + + + J4 + + J5 + J6 + + J1 + J2 J3 + + + m1 m2 m3 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 16 + a) Zeigen Sie per Widerspruch, dass der Algorithmus Round Robin Scheduling eine approxima- + tive Güte von 2 hat. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 16 + b) BONUS: Geben Sie die Konstruktionsvorschrift für eine Instanz an, mit der der Algorithmus + Round Robin Scheduling beliebig nah an eine Güte von 2 kommt, also A(I) ≈ 2 · OPT (I). + Nutzen Sie dafür mindestens 3 Maschinen und 7 Jobs, womit eine Rate von A(I) = 53 OPT + zu erreichen ist. Geben sie die optimale Makespan an und die, die der Algorithmus bei Ihrer + Instanz erreicht. Für eine Güte von 53 können bis zu 3 Punkten erreicht werden, bei einer + Güte von 47 sind bis zu 5 Punkte zu erreichen. Geben Sie eine Konstruktionsvorschrift an, + bei denen bei einer beliebigen, festen Anzahl von Maschinen m eine Güte von 2 − m1 erreicht + wird, können Sie die vollen 10 Punkte erreichen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 16 + Aufgabe 5 KOMPLEXITÄT: ETH (5+5 Punkte) + + a) Wir betrachten folgendes Problem: + Problem: DominatingSet + Eingabe: Ein ungerichteter Graph G = (V, E) mit V = 1, . . . , |V | und eine Ganzzahl k ≥ 1. + Entscheide: Gibt es eine Menge M ⊆ V der Kardinalität |M| = k derart, dass jeder Knoten + v ∈ V entweder in M ist oder zu einem Knoten in M benachbart ist. + Wir untersuchen nun folgende Reduktion von 3-SAT auf DominatingSet: + 1. Seien C1 , . . . ,Cm die Klauseln der 3-SAT-Formel + 2. Seien x1 , . . . , xn die Variablen der 3-SAT-Formel + 3. Erzeuge die Knoten xi , xi und di für jede Variable xi + 4. Erzeuge die Kanten {xi , xi }, {xi , di } und {di xi } für jede Variable xi + 5. Für jede Klausel C j erzeuge einen Knoten C j + 6. Für jedes Literal ℓ einer Klausel C j erzeuge die Kanten {C j , ℓ} + Welche unteren Schranken für die Laufzeit folgen damit für DominatingSet unter der ETH + in Bezug auf + (i) die Anzahl der Knoten |V |? + (ii) die Anzahl der Kanten |E|? + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 12 von 16 + b) Wir betrachten folgendes Problem: + Problem: GridTiling + Eingabe: Zwei ganze Zahlen k und j und für jedes i ∈ [k] und j ∈ [k] eine Menge Si, j ⊆ [n]2 . + Entscheide: Gibt es (ei, j )(i, j)∈[k]2 derart, dass + 1. für alle (i, j) ∈ [k]2 gilt ei, j ∈ Si, j ; + 2. für alle i ∈ [k] und j ∈ [k − 1] gilt für ei, j = (a, b) und ei, j+1 = (a′ , b′ ), dass a = a′ (dh. die + ausgewählten Tupel müssen in der ersten Komponente übereinstimmen, wenn der erste + Index derselbe ist); und + 3. für alle i ∈ [k − 1] und j ∈ [k] gilt für ei, j = (a, b) und ei+1, j = (a′ , b′ ), dass b = b′ (dh. + die ausgewählten Tupel müssen in der zweiten Komponente übereinstimmen, wenn der + zweite Index derselbe ist) + Betrachten Sie folgende Beispielinstanz für k = 2, die Elemente der Lösung sind unterstrichen: + + + S1,1 = {(1, 1), (2, 2)} S1,2 = {(1, 2), (2, 1), (2, 2)} + + + + S2,1 = {(1, 2), (2, 1)} S2,2 = {(1, 1), (2, 2)} + + + Wir untersuchen folgende Reduktion von Clique: + 1. Sei G = (V, E) und eine Zahl k als zu untersuchende Clique Instanz gegeben + 2. Wir nehmen ohne Beschränkung der Allgemeinheit an, dass V = {1, . . . , |V |} sowie k ≥ 2 + gilt und dass keine isolierten Knoten existieren. + 3. Wir übernehmen den Wert von k + 4. Für jedes Paar (i, j) ∈ [k]2 definieren wir + ( + { (a, a) | a ∈ V } if i = j + Si, j := + { (a, b) | a ̸= b, {a, b} ∈ E } if i ̸= j + Welche unteren Schranken für die Laufzeit folgen damit für Grid Tiling (mit der aus der + Vorlesung bekannten Reduktion von Clique auf 3-SAT) in Hinblick auf + (i) der Gesamtanzahl X := ∑(i, j)∈[k]2 |Si, j | der Tupel in den Mengen; sowie + (ii) der maximalen Anzahl Y := max(i, j)∈[k]2 |Si, j | an Tupeln in einer der Mengen? + Hinweis: Aus der Reduktion in der Vorlesung wissen wir, dass Clique selbst dann noch NP- + schwer ist, wenn k ∈ Θ(|V |) gilt. Sie dürfen also annehmen, dass es eine (globale) Konstante + c > 0 gibt, sodass c · n ≤ k ≤ n. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 13 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 14 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 15 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 16 von 16 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen + + 14. Juli 2023 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2023 + Name: Matrikel-Nr.: + + + Hinweise: + + • Bearbeiten Sie alle Aufgaben. + • Sie haben 10 Minuten Einlesezeit. Danach haben Sie 120 Minuten Zeit für die Klausurbearbeitung. + • Erlaubte Hilfsmittel: Ein doppelseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein + Rot, kein Grün, kein Bleistift). + • Einsichtnahme: Nach Absprache. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 11 + Aufgabe 1 RUCKSACKPROBLEM (3+2+5 Punkte) + Wir betrachten die Anwendung der Approximationsheuristiken für das Rucksackproblem aus der + Vorlesung. + Gegeben sei dafür folgende Instanz des Rucksackproblems mit Kapazität B = 16. Items sind in + der Form (pi , wi ) angegeben. + I = [(1, 4), (1, 1), (1, 3), (11, 13), (3, 3), (5, 6), (1, 5), 16] + a) Wenden Sie den normalen Greedy-Algorithmus auf obige Instanz an. Begründen Sie kurz, + warum die gewählten Items ausgewählt werden. Was ist der Lösungswert? + + + + + b) Wenden Sie nun den ModifiedGreedy-Algorithmus an. Wie ändert sich die Lösung? + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 11 + c) Was ist die Güte des ModifiedGreedy-Algorithmus? Geben Sie eine kurze Idee an, wie man + diesen Ansatz modifizieren kann, um die Güte auf 32 oder 43 zu verbessern. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 11 + Aufgabe 2 L ONGEST PATH (3+3+4 Punkte) + Betrachten Sie das folgende Problem: + Problem: L ONGEST PATH + Eingabe: Eine ungerichteter Graph G = (V, E) und eine Zahl k. + Entscheide: Gibt es einen Pfad in G, der mindestens Länge k hat? + + Zeigen Sie auf verschiedene Weisen, dass das Problem in NP liegt: + + a) Geben Sie ein Zertifikat für L ONGEST PATH an. + b) Beschreiben Sie einen Verifizierer für L ONGEST PATH und geben Sie dessen Laufzeit konkret + an. + c) Beschreiben Sie eine nicht-deterministische Turing-Maschine (in Worten), die L ONGEST PATH + löst. Geben Sie auch hier die Laufzeit konkret an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 11 + Aufgabe 3 A PPROXIMATIVE A LGORITHMEN (6+4 Punkte) + Das Problem M AX -3-S AT sei folgendermaßen definiert: + Problem: M AX -3-S AT + Eingabe: Eine Formel φ in konjunktiver Normalform, wobei jede Klausel drei Literale enthält. + Ausgabe: Eine Belegung β der Variablen, die die Anzahl v(β ) der erfüllten Klauseln maximiert. + + Betrachten Sie folgenden Algorithmus A: + + • Sei β0 die Belegung, die alle Variablen auf false setzt. + • Sei β1 die Belegung, die alle Variablen auf true setzt. + • Falls v(β0 ) ≥ v(β1 ), gib β0 zurück. + • Sonst gib β1 zurück. + + a) Zeigen Sie, dass obiger Algorithmus Güte 2 hat, d.h. A(φ ) ≥ 12 OPT (φ ) gilt für alle Eingaben + φ. + b) Geben Sie eine Formel an, bei der der Algorithmus eine Belegung ausgibt, die genau die Hälfte + der maximal erfüllbaren Klauseln erfüllt. Begründen Sie kurz, warum Ihre Formel geeignet ist. + Hinweis: Sie brauchen nur höchstens zwei Klauseln und sechs Variablen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 11 + Aufgabe 4 B EWEIS ZUR VORLESUNG (10 Punkte) + Betrachten Sie folgendes Problem aus der Vorlesung: + + Problem: k-C LIQUE + Eingabe: Ein ungerichteter Graph G = (V, E) und eine Zahl k ≥ 1. Eine Clique ist eine Teil- + menge C ⊆ V mit {u, v} ∈ E für alle u, v ∈ C mit u ̸= v. + Entscheide: Hat der gegebene Graph G eine Clique C ⊆ V mit mindestens k Knoten (d.h. mit + |C| ≥ k)? + + Beweisen Sie: Das Problem k-C LIQUE ist NP-vollständig. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 11 + Aufgabe 5 NP UND NP-VOLLSTÄNDIGKEIT (4+4+7 Punkte) + + + a) Beweisen Sie die NP-Vollständigkeit von S UBSET S UM, bei der jede Itemgröße jeweils durch + 3 oder durch 7 teilbar ist. + b) Beweisen Sie die NP-Vollständigkeit von 3-C OLOR, wobei jeder Knoten im Graph mindes- + tens Grad 3 hat. + c) Zeigen Sie: Für kein α > 1 gibt es einen approximativen Algorithmus mit Güte α (d.h. mit + Zielfunktionswert ≤ α · OPT) für TSP, außer P = NP (Denken Sie an Wiliam Rowan Hamil- + ton). + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 11 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 11 + Aufgabe 6 U NTERE S CHRANKEN UND ETH (7+8 Punkte) + Betrachten Sie folgende Reduktionen: + Problem: H ITTING S ET + Eingabe: Eine Menge U sowie r Teilmengen F1 , . . . , Fr ⊆ U und eine Zahl k. + Entscheide: Gibt es eine Menge S ⊆ U mit |S| ≤ k und S ∩ Fi ̸= 0/ für alle i ∈ [r]? + + a) 3-SAT ⪯ H ITTING S ET: Seien v1 , . . . , vn die Variablen der Formel und C1 , . . . ,Cm die Klau- + seln. Setze U = { v1 , v1 , . . . , vn , vn } und k = n. Für jede Variable vi definiere eine Menge + Fi = {vi , vi } und für jede Klausel C j definiere eine Menge Fn+ j = C j + Welche unteren Schranken ergeben sich durch obige Reduktion für die Laufzeit von Algorith- + men für H ITTING S ET unter Annahme der ETH in Abhängigkeit von + (i) r? + (ii) |U|? + Beweisen Sie die entsprechenden unteren Schranken! + Hinweis: Sie dürfen – wie in den Hausaufgaben zur ETH – Faktoren, die polynomiell in der + Eingabekodierung sind, vernachlässigen. + b) 3-S AT ⪯ S UBSET S UM: Seien v1 , . . . , vn die Variablen der Formel und C1 , . . . ,Cm die Klauseln. + Für jede Variable vi erzeuge zwei Items ai und bi , mit den Größen + + s(ai ) = 10i−1 + ∑ 10n+ j−1 und s(bi ) = 10i−1 + ∑ 10n+ j−1 . + j∈[m] j∈[m] + xi ∈C j xi ∈C j + + Zusätzlich erzeuge zwei Items c j und d j für jede Klausel C j mit Größen + + s(c j ) = s(d j ) = 10n+ j−1 . + + Der Zielwert sei + B = ∑ 3 · 10n+ j−1 + ∑ 10i−1 . +  + j∈[m] i∈[n] + + Welche unteren Schranken ergeben sich durch obige Reduktion für die Laufzeit von Algorith- + men für S UBSET S UM unter Annahme der ETH in Abhängigkeit von + (i) der Anzahl der unterschiedlichen Itemgrößen d? + (ii) der binären Kodierungslänge L der größten Itemgröße ∆ (d.h. L = log(∆))? + Beweisen Sie die entsprechenden unteren Schranken! + Hinweis: Sie dürfen auch hier – wie in den Hausaufgaben zur ETH – Faktoren, die polynomiell + in der Eingabekodierung sind, vernachlässigen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 11 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 11 + Viel Erfolg! + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 11 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen + + 19. Oktober 2023 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2023 + Name: Matrikel-Nr.: + + +Hinweise: + +• Bearbeiten Sie alle Aufgaben. +• Sie haben 120 Minuten Zeit für die Klausurbearbeitung. +• Erlaubte Hilfsmittel: Ein einseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein Rot, + kein Grün, kein Bleistift, kein Tintenkiller, kein Tipp-Ex). +• Schreiben Sie auf jedes Blatt ihren Namen. +• Wenn Sie Schmierzettel verwenden um eine Aufgabe zu bearbeiten, verweisen Sie in der Aufgabe auf + diese. +• Einsichtnahme: 2.11.23, 13:30-14:30 Uhr. +• Die Klausur hat 6 Aufgaben auf 13 Seiten – überprüfen Sie bitte vor der Bearbeitung, dass Ihre Klausur + vollständig ist. + Aufgabe 1 C HRISTOFIDES ’ A LGORITHMUS (3+7 Punkte) + Gegeben sei folgender Graph G = (V, E) : + + d + + 2 + 3 + + 3 + a c + + + 1 + 2 + + b + + a) Geben Sie an, was Christofides’ Algorithmus berechnet, und welche approximative Güte er + hat. Welche Eigenschaft müssen die Distanzen neben der Symmetrie noch erfüllen, damit + Christofides’ Algorithmus korrekt arbeitet. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 19 + b) Wenden Sie den Algorithmus von Christofides auf den Graphen G mit Startknoten a an. Ge- + ben Sie dabei alle Graphen an, die in Zwischenschritten entstehen. Erwähnen Sie ebenfalls + Schritte, die bei dieser Anwendung zu keiner Änderung führen, die der Algorithmus aber + überprüfen muss. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 19 + Aufgabe 2 D OMINATING S ET (3+3+4 Punkte) + Betrachten Sie das folgende Problem: + Problem: D OMINATING S ET + Eingabe: Ein ungerichteter Graph G = (V, E) (wobei V = {1, . . . , n} für eine Zahl n ∈ N≥1 ) + und eine Zahl k ∈ N0 . + Entscheide: Gibt es eine Teilmenge D ⊆ V der Knoten mit |D| ≤ k, sodass jeder Knoten v ∈ V + in D enthalten ist oder benachbart zu mindestens einem der Knoten in D ist? + + Zeigen Sie auf verschiedene Weisen, dass das Problem in NP liegt: + a) Geben Sie ein Zertifikat für D OMINATING S ET an. + b) Beschreiben Sie einen polynomiellen Verifizierer für D OMINATING S ET und schätzen Sie die + Laufzeit in O-Notation konkret ab. + c) Beschreiben Sie (in Worten) eine polynomielle, nicht-deterministische Turing-Maschine, die + D OMINATING S ET löst. Schätzen Sie auch hier die Laufzeit in O-Notation konkret ab. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 19 + Aufgabe 3 A PPROXIMATIVE A LGORITHMEN (6+4 Punkte) + Das Problem M IN -E DGE -C OVER sei folgendermaßen definiert: + Problem: M IN -E DGE -C OVER + Eingabe: Ein zusammenhängender, ungerichteter Graph G = (V, E). + Ausgabe: Eine Teilmenge der Kanten C ⊆ E mit minimaler Kardinalität, sodass jeder Knoten + v ∈ V zu mindestens einer Kante aus C inzident ist. + + Betrachten Sie folgenden Algorithmus A: + + • Gehe (in beliebiger Reihenfolge) alle Knoten durch: + • Falls der aktuelle Knoten v noch nicht abgedeckt ist, wähle zufällig eine der zu v inzidenten + Kanten und füge sie zu C hinzu. + • Gib anschließend die entstehende Menge C zurück. + + a) Zeigen Sie, dass obiger Algorithmus Güte 2 hat, d.h. A(G) ≤ 2 · OPT(G) gilt für alle Eingaben + G. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 19 + b) Sei n ∈ N≥4 gerade. Betrachten Sie den Graphen Gn , der genau n Knoten und n Kanten hat und + einen Kreis beschreibt. D.h. Gn = ({1, . . . , n}, {{i, i+1} | i ∈ {1, . . . , n−1}}∪{{n, 1}}). Wie viele + Kanten hat ein M IN -E DGE -C OVER des Graphen Gn ? Wie schlecht ist die Approximationsrate + des Algorithmus A für Gn im schlimmsten Fall? Geben Sie eine Reihenfolge der Knoten (für + Schritt 1) und eine Auswahl der Kanten (für Schritt 2) an, die zu einer möglichst schlechten + Approximationsrate führen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 19 + Aufgabe 4 B EWEIS – S CHEDULING AUF IDENTISCHEN M ASCHINEN (10 Punkte) + Betrachten Sie das Scheduling-Problem P||Cmax : + Problem: P||Cmax + Eingabe: Eine Liste L = (J1 , . . . , Jn ) von n Jobs mit Ausführungszeiten p1 , . . . , pn ∈ N, und m + identische Maschinen. + Ausgabe: Ein Schedule (Partition von J in m Teilmengen B1 , . . . , Bm ) mit minimaler maximaler + Last Cmax := max ∑ p j . + 1≤i≤m J ∈B + j i + + + Betrachten Sie den folgenden Algorithmus ListScheduling, der einen Schedule berechnet. Sei + LS(I) die Last Cmax eines Schedules des Algorithmus, und OPT(I) die Last Cmax eines optimalen + Schedules zur Eingabe I. Zeigen Sie: Für alle Eingaben I = (L, m) gilt LS(I)/OPT(I) ≤ 2 − 1/m. + + Algorithmus L IST S CHEDULING(L = (J1 , . . . , Jn ), m) + + 1 for i = 1 to m do + 2 Ei = 0; Bi = 0/ ; + 3 od + 4 for j = 1 to n do + 5 wähle Job J j aus Liste L; + 6 wähle Maschine Mi mit minimaler Last Ei ; + 7 Bi = Bi ∪ {J j }; + 8 Ei = Ei + p j ; + 9 od + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 19 + Aufgabe 5 NP UND NP-VOLLSTÄNDIGKEIT (4+5+6 Punkte) + + Problem: H AMILTONIAN PATH + Eingabe: Ein ungerichteter Graph G = (V, E) + Entscheide: Existiert ein Pfad in G, der jeden Knoten genau einmal besucht? + Problem: H AMILTONIAN C YCLE + Eingabe: Ein ungerichteter Graph G = (V, E) + Entscheide: Existiert ein Kreis in G, der jeden Knoten genau einmal besucht? + + (a) Sei L ⊆ S UBSET S UM die Menge der (positiven) SubsetSum-Instanzen, bei denen keine der + Itemgrößen eine Zweierpotenz ist. Zeigen Sie, dass L NP-vollständig ist. + (b) Geben Sie eine polynomielle Reduktion von H AMILTONIAN PATH auf H AMILTONIAN C Y- + CLE an. + + (c) Geben Sie eine polynomielle Reduktion von H AMILTONIAN C YCLE auf H AMILTONIAN - + PATH an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 19 + Aufgabe 6 U NTERE S CHRANKEN UND ETH (5+5 Punkte) + Im Folgenden bezeichne ⟨I⟩ die Kodierungslänge der Instanz. + (a) Wir betrachten folgendes Problem: + Problem: H ITTING S ET + Eingabe: Eine Menge U ′ sowie r′ Teilmengen F1′ , . . . , Fr′′ ⊆ U ′ und eine Zahl k′ . + Entscheide: Gibt es eine Menge S′ ⊆ U ′ mit |S′ | ≤ k′ und S′ ∩ Fi′ ̸= 0/ für alle i′ ∈ {1, . . . , r′ }? + + Sie dürfen nutzen, dass sich H ITTING S ET unter Annahme der ETH nicht in Zeit 2o(r ) ⟨I⟩O(1) + ′ + + + und nicht in Zeit 2o(|U |) ⟨I⟩O(1) lösen lässt. + ′ + + + Problem: S ET C OVER + Eingabe: Eine Menge U sowie r Teilmengen F1 , . . . , Fr ⊆ U und eine Zahl k. + Entscheide: Gibt es eine Menge S ⊆ {1, . . . , r} mit |S| ≤ k und i∈S Fi = U? + S + + + Welche unteren Schranken ergeben sich + (i) in Hinblick auf |U| sowie + (ii) in Hinblick auf r + für S ET C OVER unter Annahme der ETH aus den unteren Schranken für H ITTING S ET, wenn + man folgende Reduktion von SetCover auf HittingSet verwendet? + Sei (F ,U, k) eine S ET C OVER-Instanz. Setze V1 := U und V2 := {1, . . . , r}. Sei V := V1 ∪V˙ 2. + Weiter sei E = { {v, w} | v ∈ V1 , w ∈ V2 , v ∈ Fw }. Damit ist G = (V, E) ein bipartiter Graph + mit Partitionen V1 ,V2 . Setze U ′ := {1, . . . , r}. Erzeuge nun eine Menge Fw′ für jeden Knoten + w ∈ V1 , indem Fw′ := { v ∈ V2 | {v, w} ∈ E } gesetzt wird. Sei F ′ := { Fw′ | w ∈ V1 }. Mit k′ = k + iie berechnete Instanz ist nun durch (F ′ ,U ′ , k′ ) gegeben. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 12 von 19 + (b) Wir wollen uns mit einem Problem beschäftigen, das ein mächtiges Werkzeug im Bereich + der Optimierung ist: Integrale Lineare Programme (ILPs). + Problem: ILP-F EASIBILITY + Eingabe: Eine Matrix A ∈ ZM×N , eine rechte Seite b ∈ ZM . + Entscheide: Gibt es einen Vektor x ∈ ZN≥0 mit Ax ≤ b, d.h. Ai xi ≤ bi für alle i ∈ {1, . . . , M}? + Welche unteren Schranken ergeben sich + (i) in Hinblick auf M sowie + (ii) im Hinblick auf N + für ILP-F EASIBILITY unter Annahme der ETH, wenn man folgende Reduktion von 3-SAT + auf ILP-F EASIBILITY verwendet? + Sei φ eine 3-SAT Instanz mit m Klauseln (Ci )i∈m und n Variablen. Analog zu linearen Glei- + chungssystemen (Ax = b), die aus M Gleichungen bestehen, haben wir bei Ax ≤ b einen + Satz von M linearen Ungleichungen. Wir konstruieren diese wie folgt: Wir nutzen N := 2n + Variablen in unserem ILP, eine für jedes (mögliche) Literal in der SAT-Formel. Für je- + de SAT-Variable v sei xv die ILP-Variable, die zum SAT-Literal v korrespondiert, und xv̄ + die, die zum SAT-Literal v̄ korrespondiert. Für jede Klausel Ci erzeugen wir eine Unglei- + chung − ∑ℓ∈Ci xℓ ≤ −1. Weiter erzeugen wir für jede SAT-Variable v zwei Ungleichungen: + xv +xv̄ ≤ 1 und −xv −xv̄ ≤ −1. Durch das Eintragen der entsprechenden Koeffizienten in eine + erweiterte Koeffizientenmatrix (bestehend aus Matrix A und rechter Seite b) erhalten wir die + ILP-F EASIBILITY Instanz. + + + + + Viel Erfolg! + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 13 von 19 + \ No newline at end of file diff --git a/uni/aak (Copy)/alle_praesenzaufgaben.pdf b/uni/aak (Copy)/alle_praesenzaufgaben.pdf new file mode 100644 index 0000000000000000000000000000000000000000..73af44c15d9b60655408c6d4af3b25a70d43fef2 GIT binary patch literal 648498 zcmd421z43^w>B&xEl7jZqC@HK?(XjHTBLM$BOu)k(j|&?NOvh+(yi3DP;`5a@7X85 z{hj~6F0aM2o;Bkh_q@lLF{Y$)LL$`kG>mYh?=Nz0;phQ$09ykKI4&*#t+X*v-$);* z4`2knQ~=P*>6;om0+@b$QF6002G9zEezvkT1@Q2|8QU1$S7i9D0e&Z-nXLnWf)Z5F z!B`(?Zfhf?4>Sf)2yrmbF)-7y(lOC9G0`!yJ*T512mSk5?!JzczLK#Tz{MPB1~3D4 z<=|j!0{U6n*62TLLC(R}$jQ(c)P$Xp38)Q_aZGeH3@nttfxI{9rs z{cBnx!1p3ROKu`Nb4z0oej*+?0GET235SA^$U`mAH*O*`AkdD3me$3^g~o-E#@4}< 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0 HcmV?d00001 diff --git a/uni/aak (Copy)/alle_praesenzaufgaben.txt b/uni/aak (Copy)/alle_praesenzaufgaben.txt new file mode 100644 index 0000000..6f985bd --- /dev/null +++ b/uni/aak (Copy)/alle_praesenzaufgaben.txt @@ -0,0 +1,480 @@ + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 09.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 9 + + +Definition 1 (Knapsack (Entscheidungsvariante)). Gegeben ist eine Menge mit n Gegenständen +mit Gewichten w1 , . . . , wn ∈ N≥0 und Profiten p1 , . . . , pn ∈ N≥0 . Weiter sind eine Kapazität +K ∈ N≥0 und ein Zielprofit P ∈ N≥0 gegeben. Bei dem K NAPSACK Problem muss entschieden +werden, ob eine Teilmenge S ⊆ [n] existiert, sodass + + ∑ wi ≤ K und ∑ pi ≥ P + i∈S i∈S + +gilt. + +Präsenzaufgabe 9.1 (K NAPSACK in NP) +Zeigen Sie K NAPSACK ∈ NP auf zwei verschiedene Wege. + 1. Geben Sie einen nicht deterministischen Algorithmus an, der das Problem in polynomi- + eller Zeit löst. + 2. Geben Sie einen polynomiellen Verifizierer für das Problem an. +Wie hängen diese beiden Methoden zusammen? + +Lösung. 1. Der Algorithmus arbeitet wie folgt: Für jeden Gegenstand entscheiden wir nicht- + deterministisch, ob er in den Rucksack soll. Anschließend akzeptieren wir, falls die Kapazitäts- + und Profitbedingung erfüllt sind. + Existiert eine valide Rucksackfüllung mit genügend Profit, dann gibt es auch eine Reihe + von nichtdeterministischen Entscheidungen, die diese Füllung findet. Diese wird dann + auch vom Algorithmus akzeptiert. + Der Algorithmus macht polynomiell viele nichtdeterministische Entscheidungen und die + Überprüfung der Eigenschaften ist ebenfalls in Polynomialzeit möglich, also ist dies ein + polynomieller, nichtdeterministischer Algorithmus, der K NAPSACK löst. + 2. Unser Zertifikat ist eine Teilmenge S ⊆ [n], die die Auswahl für den Rucksack darstellen + soll. Dies kann zum Beispiel über ein Bit pro Gegenstand realisiert werden. Um dieses + Zertifikat zu verifizieren, berechnen wir ∑i∈S wi und ∑i∈S pi und vergleichen die Werte + mit K und P und akzeptieren entsprechend. + Existiert eine valide Rucksackfüllung mit genügend Profit, dann gibt es auch ein Zertifi- + kat, das diese Füllung beschreibt. Dieses wird dann auch vom Verifizierer akzeptiert. + Das Ausrechnen der Summen und das Überprüfen ist in polynomieller Zeit möglich, also + ist dies ein polynomieller Verifizierer. + In beiden Wegen wird die extra Ressource, die wir zur Verfügung haben, genutzt, um eine +Teilmenge (einen möglichen Rucksack) zu bestimmen und zu überprüfen. In der Vorlesung +wird noch genau gezeigt, dass diese beiden Wege tatsächlich äquivalent sind. + +Präsenzaufgabe 9.2 (Transitivität von Polynomialzeitreduktionen) +Zeigen Sie, dass Polynomialzeitreduktionen transitiv sind, d.h., dass für alle Entscheidungspro- +bleme L1 , L2 , L3 ⊆ Σ∗ gilt, dass + + L1 ⪯ L2 ∧ L2 ⪯ L3 =⇒ L1 ⪯ L3 . + +Lösung. Seien L1 , L2 , L3 Sprachen. Zudem sei R1 ein Algorithmus, der L1 nach L2 in O(na ) +reduziert, und R2 ein Algorithmus, der L2 nach L3 in O(nb ) reduziert. +Sei x eine Instanz von L1 der Größe n. Dann benötigt der Aufruf von R1 auf x höchstens O(na ) +Zeit. +Sei x′ die Ausgabe von R1 . Dann hat x′ eine Größe von höchstens O(na ). Diese Schranke für +die Ausgabe ergibt sich aus der Laufzeit der Reduktion. Der Aufruf von R2 auf x′ benötigt nun +höchstens O((na )b ) = O(nab ) Zeit. Damit ist die Reduktion von L1 auf L3 in polynomieller Zeit +möglich. +Die Korrektheit ergibt sich aus der Transitivität der Äquivalenz: Sei x′′ die Ausgabe von R2 auf +x′ , dann gilt + x ∈ L1 ⇐⇒ x′ ∈ L2 und x′ ∈ L2 ⇐⇒ x′′ ∈ L3 , +also auch x ∈ L1 ⇐⇒ x′′ ∈ L3 . + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 16.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 10 + + +Präsenzaufgabe 10.1 (Fragen über Fragen) +Diskutieren Sie die folgenden Aussagen (Gelten sie allgemein? Gelten sie nicht? Gelten sie +unter bestimmten Bedingungen? Welche?): + (i) Für jede Sprache, die von einer NDTM in polynomieller Zeit akzeptiert werden kann, + existiert eine DTM, die ebenfalls die Sprache in polynomieller Zeit akzeptiert. + (ii) A ist NP-schwer =⇒ A ∈ P + +Lösung. (i) Die Aussage gilt nur, wenn P = NP. Dann liegen alle Probleme in NP (von + NDTM in polynomieller Zeit lösbar) auch in P (von DTM in polynomieller Zeit lösbar). + Falls P ̸= NP, dann nein, weil sonst folgender Widerspruch auftritt: ein Problem in NP \ P + ist von einer NDTM, nicht aber von einer DTM in polynomieller Zeit lösbar. Könnte eine + DTM die NDTM in polynomieller Zeit simulieren, wäre das Problem auch in P. + (ii) Falsch. Es gibt NP-schwere Probleme, die nicht in NP liegen, z.B. das H ALTE-Problem. + Diese liegen folglich auch definitiv nicht in P. NP-schwer ist eine untere Schranke, ∈ NP + eine obere, und wenn beide erfüllt sind, ist ein Problem NP-vollständig. + + + +Definition 1 (Vertex Cover (VC)). Für einen (ungerichteten) Graphen G = (V, E) ist ein Vertex +Cover eine Menge von Knoten C ⊆ V , sodass u ∈ C oder v ∈ C für alle { u, v } ∈ E gilt, also ist +für alle Kanten mindestens ein Endpunkt in C. +Für das V ERTEX C OVER Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gegeben und +es wird gefragt, ob ein Vertex Cover mit Größe (Anzahl der Knoten) höchstens k in G existiert. + +Präsenzaufgabe 10.2 (V ERTEX C OVER ist NP-vollständig) +Zeigen Sie, dass V ERTEX C OVER NP-vollständig ist. +Hinweis: Reduzieren Sie C LIQUE auf V ERTEX C OVER. + +Lösung. V ERTEX C OVER ∈ NP ist bekannt aus der letzten Hausaufgabe. +Vorüberlegung zur Reduktion von C LIQUE auf V ERTEX C OVER: Sei eine Instanz mit G = +(V, E) und k ∈ N≥0 von C LIQUE gegeben. Wir invertieren zunächst den Graphen G und erhal- +ten damit G′ . Nun enthält G′ genau dann eine unabhängige Menge der Größe ℓ, wenn G eine +Clique der Größe ℓ enthält. Wenn ein Graph mit n Knoten eine unabhängige Menge mit min- +destens k Knoten hat, dann hat der Graph auch ein Vertex Cover mit maximal n − k Knoten + (das Komplement zur unabhängigen Menge). Unsere Instanz für V ERTEX C OVER ist also G′ +und n − k. (k ≤ n können wir annehmen) +Betrachte nun folgende Reduktion: +Sei eine Instanz mit G = (V, E) und k ∈ N≥0 von C LIQUE gegeben. Invertiere den Graphen G +und erhalte damit G′ . Gebe nun G′ und n − k als Instanz von V ERTEX C OVER zurück. +Korrektheit: +⇒ (Falls (G, k) eine Ja-Instanz von C LIQUE ist, dann ist (G′ , n − k) eine Ja-Instanz von V ER - +TEX C OVER .): + +G enthalte eine Clique C der Größe ≥ k. Dann ist V ′ := V \C ein Vertex Cover von G′ der Größe +≤ n − k. Da |C| ≥ k, gilt |V ′ | ≤ n − k. Da C eine Clique ist, also alle Knoten aus C mit allen +anderen Knoten aus C verbunden sind, gibt es im invertierten Graphen G′ keine Kante zwischen +Knoten aus C. Also gilt für jede Kante {u, v} ∈ E ′ , dass u ∈ / C und somit u ∈ V ′ oder + / C oder v ∈ +v ∈ V ′. +⇐ (Falls (G′ , n − k) eine Ja-Instanz von V ERTEX C OVER ist, dann ist (G, k) eine Ja-Instanz von +C LIQUE.): +G′ enthalte ein Vertex Cover V ′ der Größe ≤ n−k. Dann ist C := V \V ′ eine Clique der Größe ≥ +k in G. Da |V ′ | ≤ n − k, gilt |C| ≥ k. Für jede Menge {u, v} ⊆ C gilt, dass {u, v} ∈ + / E ′ . Ansonsten +wäre V ′ kein korrektes Vertex Cover. Da G′ der invertierte Graph von G ist, gilt somit {u, v} ∈ E +für jede Menge {u, v} ⊆ C. +Laufzeit: Das Invertieren des Graphen kann naiv in O(|V |2 · |E|) Zeit, also in polynomieller +Zeit, erledigt werden. +Also existiert eine polynomielle Reduktion von C LIQUE auf V ERTEX C OVER. Da C LIQUE +NP-vollständig ist und V ERTEX C OVER ∈ NP gilt, ist somit V ERTEX C OVER NP-vollständig. + + +Präsenzaufgabe 10.3 (Turingmaschinen) + n +Entwerfen Sie eine Turingmaschine für die Sprache L = { 02 | n ∈ N } über dem Alphabet +Σ = {0} und geben Sie die Laufzeit Ihrer Turingmaschine an. Begründen Sie die Korrektheit +und die Laufzeit Ihrer Turingmaschine. Diskutieren Sie die Laufzeit Ihrer Turingmaschine im +Vergleich zu einem Algorithmus (mit RAM) für diese Sprache. + +Lösung. Die Turingmaschine sieht folgendermaßen aus: + (1) Falls genau eine 0 auf dem Band steht, akzeptiere. + (2) Bewege den Kopf von links nach rechts über die Eingabe und ersetze jede zweite 0 durch + ein neues Symbol x. + (3) Falls das letzte Symbol der Eingabe nun eine 0 ist, verwerfe. + (4) Gehe zu Schritt (1). +Korrektheit: Die Sprache L besteht aus allen Wörtern, die nur Nullen enthalten und deren Länge +eine Zweierpotenz ist. Eine Zahl ist genau dann eine Zweierpotenz, wenn sie nach wiederholtem +Teilen durch 2 irgendwann eine 1 ergibt. Genau das überprüft die Turingmaschine. Wenn bei +diesem wiederholten Teilen eine ungerade Zahl auftritt, ist die Zahl keine Zweierpotenz. In +diesem Fall verwirft die Turingmaschine die Eingabe (siehe Schritt (3)). +Laufzeit: Die angegebene Turingmaschine hat eine Laufzeit von O(n log(n)), wobei n hier die +Länge der Eingabe bezeichnet. Schritt (1) und Schritt (2) benötigen jeweils eine Laufzeit von + O(n). Schritt (3) geht in konstanter Laufzeit O(1). Schritt (4) wiederholt die Schritte (1) bis (3) +bis nur noch eine 0 auf dem Band steht oder vorher abgebrochen wird. Da in jedem Durchlauf +der Schritte (1) bis (3) die Hälfte der Nullen (abgerundet) durch x Symbole ersetzt wird, steht +nach log n Durchläufen nur noch eine Null auf dem Band und es wird nach Schritt (1) akzeptiert +oder es steht nach ⌊log n⌋ − 1 Durchläufen eine ungerade Anzahl von Nullen auf dem Band und +es wird nach Schritt (3) verworfen. Also gibt es maximal log n + 1 Durchläufe. +Diskussion: Ein Algorithmus mit RAM könnte folgendermaßen aussehen: + 1. Zähle die Buchstaben und erhalte Anzahl m. + 2. Falls log m ganzzahlig ist, akzeptiere. Sonst verwerfe. +Dieser Algorithmus hat eine Laufzeit von O(n), wobei n hier die Länge der Eingabe bezeichnet. +Ein Algorithmus mit RAM kann also den zusätzlichen Speicher nutzen, um wesentlich schneller +eine Aufgabe zu erledigen, im Vergleich zu einem Algorithmus, der auf einer Turingmaschine +läuft. + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 23.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 11 + + +Präsenzaufgabe 11.1 (Fragen über Fragen) +Diskutieren Sie nachfolgende Aussagen. Sind diese wahr oder falsch? +(a) Für L ∈ NP gilt, aus L ⪯ 3-SAT folgt, dass L NP-vollständig ist. +(b) Gilt für eine Sprache L, dass sowohl L ⪯ 3-SAT als auch 3-SAT ⪯ L, dann ist L NP-vollständig. +(c) Sei L NP-vollständig. Dann gilt L ∈ P genau dann, wenn P = NP. +(d) Es ist möglich, dass 3-SAT ∈ P und C LIQUE ∈ + / P. + +Lösung. + / Offensichtlich gilt L ∈ NP. Aus L ⪯ 3-SAT folgt nur, +(a) Nein, ein Gegenbeispiel ist L = 0. + dass sich L auf 3-SAT reduzieren lässt, nicht anders herum. +(b) Das ist die Definition von NP-vollständig. +(c) Korrekt, da sich jedes Problem auf L reduzieren lässt nach Def. NP-vollständig (und offen- + sichtlich P ⊆ NP). +(d) Nein, weil 3-SAT NP-vollständig ist, lässt sich C LIQUE (liegt bekannterweise in NP, ist + sogar NP-vollständig) auf 3-SAT reduzieren. + + + +Präsenzaufgabe 11.2 (NP-Vollständigkeit) +In dem Problem C LIQUE -N OMEMBER ist ein Graph G = (V, E), ein Knoten v ∈ V sowie eine +Zahl k ∈ N gegeben und es soll entschieden werden, ob es eine Clique mit k Knoten gibt, die +den Knoten v nicht enthält. +Zeigen Sie, dass C LIQUE -N OMEMBER NP-vollständig ist, indem Sie eine Reduktion von C LI - +QUE auf C LIQUE -N OMEMBER angeben. Sie können davon ausgehen, dass C LIQUE -N OMEMBER ∈ +NP ist. + +Lösung. Clique-Nomember ist NP-schwer: +Sei (G = (V, E), k) eine Eingabe für das Clique Problem. +Idee: Füge einen neuen isolierten Knoten v zu G hinzu. +Wir geben zuerst die Abbildung an: Sei v ̸∈ V . Wir definieren die Eingabe für das Problem +C LIQUE -N OMEMBER als den Graphen G′ := (V ∪ {v}, E), den Knoten v als Knoten, der nicht + benutzt werden darf, und k als gesuchte Cliquengröße. (G′ , v, k) ist eine gültige Eingabe für +C LIQUE -N OMEMBER und in polynomieller Zeit berechenbar. +Wir zeigen nun: G hat eine k-Clique genau dann, wenn G′ eine k-Clique ohne v beinhaltet. +⇒: +Sei (G, k) eine Ja-Instanz des C LIQUE Problems. Dann enthält G eine k-Clique C. Dann ist C +auch eine k-Clique in G′ , da bei der Abbildung keine Kanten entfernt wurden. Da v ̸∈ V , enthält +C den Knoten v nicht. Damit ist C eine k-Clique in G ohne den Knoten v. (G′ , v, k) ist also eine +Ja-Instanz des C LIQUE -N OMEMBER Problems. +⇐: +Sei (G′ , v, k) eine Ja-Instanz des C LIQUE -N OMEMBER Problems. Dann enthält G′ eine k-Clique +C mit v ̸∈ C. Da bei der Abbildung an dem Graphen nichts geändert wurde, außer der Knoten v +hinzugefügt, muss C auch eine k-Clique in G sein. Damit ist (G, k) eine Ja-Instanz des C LIQUE +Problems. +Also existiert eine polynomielle Reduktion von C LIQUE auf C LIQUE -N OMEMBER. Da C LI - +QUE NP-vollständig ist und C LIQUE -N OMEMBER ∈ NP nach Aufgabenstellung gilt, ist somit +C LIQUE -N OMEMBER NP-vollständig. + + + +Präsenzaufgabe 11.3 (NP-Vollständigkeit) +Für das Problem S UBSET S UM C ARDINALITY seien n ganze Zahlen c1 , . . . , cn ∈ N>0 (wobei n +gerade sei) sowie eine Zahl K ∈ N gegeben. Es soll entschieden werden, ob es eine Teilmenge +S ⊂ {1, . . . , n} gibt mit |S| = n/2 und ∑i∈S ci = K. +Zeigen Sie, dass das Problem S UBSET S UM C ARDINALITY NP-vollständig ist. Sie können +davon ausgehen, dass S UBSET S UM C ARDINALITY ∈ NP ist. + +Lösung. Subset Sum Cardinality ist NP-schwer: +S UBSET S UM Problem: Gegeben: n ganze Zahlen c1 , . . . , cn ∈ N>0 und Zahl K ∈ N. Frage: Gibt +es eine Teilmenge S ⊆ {0, . . . , n} mit ∑i∈S ci = K? +Reduktion: Sei I = (c1 , . . . , cn , K) eine Eingabe für das S UBSET S UM Problem. Definiere c′i := +ci + 1 für i ≤ n und c′i := 1 für i ∈ {n + 1, . . . , 2n}. Wir definieren die Eingabe für das Problem +S UBSET S UM C ARDINALITY als I ′ = (c′1 , . . . , c′2n , K + n). +Wir zeigen nun: Es gibt eine Lösung für I genau dann, wenn es eine Lösung für I ′ gibt. +⇒: +Sei I eine Ja-Instanz des S UBSET S UM Problems. Dann gibt es eine Menge S ⊆ {1, . . . , n} mit +∑i∈S ci = K. Es gilt also ∑i∈S c′i = ∑i∈S (ci + 1) = K + |S|. Sei nun S′ = {n + 1, . . . , 2n − |S|}. Es +gilt |S ∪ S′ | = n und ∑i∈S∪S′ c′i = ∑i∈S c′i + ∑i∈S′ c′i = ∑i∈S (ci + 1) + ∑i∈S′ 1 = K + |S| + (n − |S|) = +K + n. S ∪ S′ ist also eine Lösung für I ′ und I ′ damit eine Ja-Instanz. +⇐: +Es sei I ′ eine Ja-Instanz. Dann gibt es also eine Menge S′ ⊆ {1, . . . , 2n} mit ∑i∈S′ c′i = K + n und +|S′ | = n. Sei S := S′ ∩{1, . . . , n}. Es gilt ∑i∈S′ c′i − ∑i∈S c′i = n−|S|, da die Items in {n+1, . . . , 2n} +das Gewicht 1 haben, und genau n − |S| Items übrig sind. Damit gilt ∑i∈S (ci + 1) = ∑i∈S c′i = +K + n − (n − |S|) = K + |S|, woraus folgt, dass ∑i∈S ci = K. Damit ist S eine Lösung für I und +somit I eine Ja-Instanz. + Laufzeit: Die Reduktion kann naiv in einer Zeit von O(n) implementiert werden. Das ist eine +polynomielle Laufzeit. +Also existiert eine polynomielle Reduktion von S UBSET S UM auf S UBSET S UM C ARDINALI - +TY . Da nach Skript S UBSET S UM NP-vollständig ist und S UBSET S UM C ARDINALITY ∈ NP +nach Aufgabenstellung gilt, ist somit S UBSET S UM C ARDINALITY NP-vollständig. + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 30.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 12 + + +Präsenzaufgabe 12.1 +Im Folgenden sollen Sie herausfinden (und begründen), welche Lower Bounds sich über ETH +und die Reduktionen aus der Vorlesung ergeben. Nutzen Sie hierfür auch das Sparsification +Lemma. +Welche Schranken für die Laufzeit in Abhängigkeit von den natürlichen Parametern (Anzahl +Knoten, Anzahl Kanten, Anzahl Items, Anzahl unterschiedlicher Items, Anzahl Tripel, Kardi- +nalität der Grundmenge, Anzahl unterschiedlicher Tripel, Anzahl unterschiedlicher auftretender +Zahlen, größte auftretende Zahl, . . . ) ergeben sich jeweils aus folgenden (zum Großteil aus der +Vorlesung bekannten) Reduktionen (Laufzeitfaktoren, die polynomiell in der Eingabekodierung +sind, dürfen vernachlässigt werden)? +Schätzen Sie dazu zunächst die verschiedenen Parameter der resultierenden Instanz mithilfe der +Parameter der eingegebenen Instanz geeignet ab. + 1. 3-SAT ⪯ k-C LIQUE + 2. k-C LIQUE ⪯ k-I NDEPENDENT S ET (Problemdefinition für die Antwort nicht relevant; in + der Reduktion wird der Graph G = (V, E) auf den Komplementgraph (V, E) abgebildet) + 3. SAT ⪯ 3-D IMENSIONAL M ATCHING (Hinweis: 3-SAT ⊆ SAT) + 4. 3-D IMENSIONAL M ATCHING ⪯ 3-E XACT C OVER + 5. 3-E XACT C OVER ⪯ S UBSET S UM + +Lösung. Angenommen ETH gilt. Das Sparsification Lemma impliziert, dass man 3-SAT nicht +in 2o(m) · |I|O(1) lösen kann. + + 1. In der Reduktion werden |V | = O(m) Knoten und |E| = O(m2 ) Kanten erstellt und k = m. + Außerdem gilt n ≤ 3m ∈ O(m). + • Lower bound bzgl. der Anzahl an Knoten: 2o(|V |) · |I|O(1) + Angenommen, es existiert ein Algorithmus, der k-Clique in 2o(|V |) ·|I|O(1) löst. Dann + gibt es auch einen Algorithmus, der 3-SAT in 2o(m) · |I|O(1) löst (Reduktion auf k- + Clique + k-Cliquen Algorithmus). Nach dem Sparsification Lemma gibt es dies nur, + wenn die ETH nicht gilt. + √ + • Lower bound bzgl. der Anzahl an Kanten: 2o( |E|) · |I|O(1) √ + Angenommen, es existiert ein Algorithmus, der k-Clique in 2o( |E|) · |I|O(1) löst. + Da |E| ∈ O(m2 ), gibt es dann auch einen Algorithmus, der 3-SAT in 2o(m) · |I|O(1) + löst (Reduktion auf k-Clique + k-Cliquen Algorithmus). Nach dem Sparsification + Lemma gibt es dies nur, wenn die ETH nicht gilt. + • Lower bound bzgl. k: 2o(k) · |I|O(1) + Angenommen, es existiert ein Algorithmus, der k-Clique in 2o(k) · |I|O(1) löst. Dann + gibt es auch einen Algorithmus, der 3-SAT in 2o(m) · |I|O(1) löst (Reduktion auf k- + Clique + k-Cliquen Algorithmus). Nach dem Sparsification Lemma gibt es dies nur, + wenn die ETH nicht gilt. +2. In der Reduktion werden genau |V | Knoten und maximal |E| = O(|V |2 ) Kanten erstellt. + • Lower bound bzgl. der Anzahl an Knoten: 2o(|V |) · |I|O(1) + Angenommen, es existiert ein Algorithmus, der k-Independent-Set in 2o(|V |) · |I|O(1) + löst. Dann gibt es auch einen Algorithmus, der k-Clique in 2o(|V |) · |I|O(1) löst (Re- + duktion auf k-Independent-Set + k-Independent-Set Algorithmus). Nach Aufgaben- + teil 1, gibt es diesen nur, wenn die ETH nicht gilt. + √ + o( |E|) + • Lower bound bzgl. der Anzahl an Kanten: 2 · |I|O(1) √ + Angenommen, es existiert ein Algorithmus, der k-Independen-Set in 2o( |E|) ·|I|O(1) + löst. Da |E| ≤ |V |2 , gibt es dann auch einen Algorithmus, der k-Clique in 2o(|V |) · + |I|O(1) löst (Reduktion auf k-Independent-Set + k-Independent-Set Algorithmus). + Nach Aufgabenteil 1, gibt es diesen nur, wenn die ETH nicht gilt. +3. In der Reduktion werden Mengen mit |U| = |V | = |W | = O(mn) Elementen erstellt und + |T | = O(m2 n2 ) Tupel. Es gilt n ≤ 3m ∈ O(m) für 3-SAT. + √ + • Lower bound bzgl. der Anzahl an Elementen: 2o( |V |) · |I|O(1) √ + Angenommen, es existiert ein Algorithmus, der 3-Dimensional-Matching in 2o( |V |) · + |I|O(1) löst. Da |V | ∈ O(mn) ≤ O(m2 ), gibt es dann auch einen Algorithmus, der + 3-SAT in 2o(m) · |I|O(1) löst (Reduktion von SAT auf 3-Dimensional-Matching an- + gewandt auf 3-SAT Instanzen + 3-Dimensional-Matching Algorithmus). Nach dem + Sparsification Lemma geht dies nur, wenn die ETH nicht gilt. + √ + 4 + • Lower bound bzgl. der Kardinalität von T : 2o( |T |) · |I|O(1) √ + 4 + Angenommen, es existiert ein Algorithmus, der 3-Dimensional-Matching in 2o( |T |) · + |I|O(1) löst. Da |T | ∈ O(m2 n2 ) ≤ O(m4 ), gibt es dann auch einen Algorithmus, der + 3-SAT in 2o(m) · |I|O(1) löst (Reduktion von SAT auf 3-Dimensional-Matching an- + gewandt auf 3-SAT Instanzen + 3-Dimensional-Matching Algorithmus). Nach dem + Sparsification Lemma geht dies nur, wenn die ETH nicht gilt. +4. Das 3-dimensionale Matching ist ein Spezialfall vom 3-Exact Cover. Daher gelten die + Lower Bounds von 3-dimensionalem Matching auch für 3-Exact Cover. Da |U| = 3m ∈ + O(|V |), ergibt sich, dass es für 3-Exact Cover unter der ETH keinen Algorithmus √ mit + √ + o( m) O(1) o( |V |) + Laufzeit 2 · |I| geben kann, da sonst 3-dimensionales Matching in 2 · + |I|O(1) lösbar wäre. Und da |F| = |T |, √ ergibt sich, dass es für 3-Exact Cover unter der ETH + o( 4 |F|) + keinen Algorithmus + √ mit Laufzeit 2 · |I|O(1) geben kann, da sonst 3-dimensionales + 4 + Matching in 2o( |T |) · |I|O(1) lösbar wäre. + √ + 4 +5. In der Reduktion wird n = |F| gesetzt. Lower bound bzgl. n: 2o( n)√· |I|O(1) + 4 + Angenommen, es existiert ein Algorithmus, der SubsetSum in 2o( n) · |I|O(1) löst. Dann + √ + 4 + |F|) +gibt es auch einen Algorithmus, der 3-Exact Cover in 2o( · |I|O(1) löst. Nach Aufga- +benteil 4, gibt es diesen nur, wenn die ETH nicht gilt. + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 07.07.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 13 + + +Präsenzaufgabe 13.1 (Vertex Cover) +Ein Vertex Cover in einem Graphen G = (V, E) ist eine Menge von Knoten C ⊆ V , so dass für +jede Kante {v, w} ∈ E gilt, dass v ∈ C oder w ∈ C. Beim Optimierungsproblem Minimum Vertex +Cover soll die kardinalitätskleinste Menge C ⊆ V gefunden werden, die ein Vertex Cover von +G ist. +Gegeben ist der folgende approximative Algorithmus mit Güte 2 für VertexCover: +Idee: Wähle eine beliebige Kante. Wenn einer der beiden Endknoten bereits in C ist, verwirf +diese Kante. Sonst, füge beide Endknoten zur Menge C hinzu. Mache dies für jede Kante. + +Algorithmus 2A PPROX CV(G = (V, E)) + + 1 C = {} + 2 foreach {u, v} ∈ E do + 3 if u ̸∈ C and v ̸∈ C then + 4 C = C ∪ {u, v} + 5 fi + 6 od + 7 return C + +Beweisen Sie die Güte. + +Lösung. Beweis: Sei C∗ ein minimales Vertex Cover. Wir zeigen |C| ≤ 2|C∗ |. +Sei A die Menge der Kanten, für die beide Endknoten zu C hinzugefügt wurden. Es gilt |C| = +2|A|, da beide Knoten jeder ausgewählten Kante zu C hinzugefügt wurden und alle anderen +Kanten nicht weiter betrachtet werden. Damit folgt auch, dass keine zwei Kanten aus A einen +Knoten gemeinsam haben können. Sei nun C∗ ein minimales Vertex Cover. C∗ enthält nach +Definition mindestens einen Knoten jeder Kante, also insbesondere einen Knoten jeder Kante +aus A. Da keine zwei Kanten aus A vom gleichen Knoten aus C∗ überdeckt werden können, gilt +|C∗ | ≥ |A|. Es folgt |C| = 2|A| ≤ 2|C∗ |. Die Lösung C von 2ApproxCV ist also maximal doppelt +so groß wie die optimale Lösung C∗ . Damit berechnet der Algorithmus eine 2-Approximation. + Präsenzaufgabe 13.2 (TSP) +In der Vorlesung haben Sie zwei approximative Algorithmen für das TSP-Problem kennen ge- +lernt. Der erste Algorithmus ∆T SP1 hat eine Approximationsrate von 2, während der zweite +Algorithmus ∆T SP2 von Christofides eine Approximationsrate von 3/2 besitzt. In dieser Übung +soll gezeigt werden, dass die zweite Rate scharf ist. D.h., dass es eine Folge von Instanzen gibt, +auf denen der Algorithmus immer näher an diese Rate heran kommt. +Wir betrachten den vollständigen Graphen Kn = (V, E) für ein n ∈ N≥6 mit n mod 4 = 2 und +die nachfolgende Gewichtsfunktion d2 : E → R+ . Zeigen Sie, dass + (a) OPT((Kn , d2 )) = n gilt und + (b) eine mögliche Ausführung von ∆TSP2 existiert, die auf Eingabe (Kn , d2 ) eine Tour der + Länge (n − 1) + n/2 berechnet. + + + 1 1 1 1 + 1 3 5 7 9 n−1 + + + 1 1 1 1 1 1 + + 1 1 1 1 + 2 4 6 8 10 n + + + +Wir definieren E2 := {{i, i + 2}|i ∈ {1, . . . , n − 2}} ∪ {{2i + 1, 2i + 2}|i ∈ {0, . . . , n/2 − 1}} und +d2 , so dass alle Kanten aus E2 auf 1 abgebildet werden und alle Kanten aus E \ E2 auf den Wert +des kürzesten Wegs zwischen diesen Knoten in dem oben abgebildeten Graphen abgebildet +werden. + +Lösung. Wichtig bei dieser Aufgabe ist der folgende Fakt: Bei beiden Algorithmen wird ir- +gendein MST genommen und dann irgendeine Reihenfolge der Knoten. Typischerweise wird +diese Reihenfolge durch bestimmte Knoten- oder Kanten-Labels festgelegt (wird zum Beispiel +der Kruskal-Algorithmus zur Konstruktion des MST genutzt, so werden dort die Kanten nach +Gewicht sortiert. Kanten gleichen Gewichts müssen also auch noch durch irgendeine Reihen- +folge festgelegt werden.) Wir gehen also davon aus, dass wir diese Reihenfolge auch wählen +können. Sollte der Algorithmus also einen “besseren” MST oder eine “bessere” Knotenreihen- +folge nutzen, passen wir die Labels dementsprechend an. +Ein möglicher MST ist durch die unten angegebene Zick-Zack-Tour gegeben oder formal durch + h [ i + ET = {{ j, j + 1}, { j + 1, j + 3}, { j + 3, j + 2}, { j + 2, j + 4}}} ∪ {{n − 1, n}}. + 1≤ j≤n−5 + j mod 4=1 + + + 1 1 1 1 + 1 3 5 7 n−3 n−1 + + + 1 1 1 1 1 1 + + 1 1 1 1 + 2 4 6 8 n−2 n + Es gilt d(T ) = n − 1 für die Kosten d(T ) des MST T , da nur Kanten mit Kosten 1 gewählt +werden und n − 1 Kanten gewählt werden müssen, um den Zusammenhang sicher zu stellen. +Da jeder Knoten im MST außer 1 und n geraden Grad hat, ist das Matching K sehr simpel, +nämlich K = {{1, n}}. Die Kosten der Kante {1, n} sind n/2 (der kürzeste Pfad von 1 zu n) +und somit gilt d(K) = n/2 für die Kosten von K. Die Rundreise R entspricht einfach dem MST +vereinigt mit der Matchingkante und hat somit Kosten: d(R) = n − 1 + n/2. +Die optimale Tour würde erst “oben” und dann “unten” entlang laufen, also von Knoten 1 zu +Knoten 3 gehen, dann zu Knoten 5 und so weiter, bis Knoten n − 1 erreicht wurde. Danach +folgen Knoten n, n − 2, n − 4 und so weiter. Diese Tour hat offensichtlich Kosten n, da wir n +Kanten mit Gewicht 1 nutzen. Somit gilt OPT((Kn , d2 )) = n und wir haben eine Rate von + + ∆T SP2 ((Kn , d2 )) n − 1 + n/2 1 + = = 3/2 − ≤ 3/2. + OPT((Kn , d2 )) n n + +Da limn→∞ 3/2 − n1 = 3/2, kommen wir der Rate von 3/2 also beliebig nahe. + \ No newline at end of file diff --git a/uni/aak/Skript.pdf b/uni/aak/Skript.pdf new file mode 100644 index 0000000..030f074 --- /dev/null +++ b/uni/aak/Skript.pdf @@ -0,0 +1,4456 @@ +%PDF-1.3 +% +1 0 obj +<< /Metadata 3 0 R /Pages 4 0 R /Type /Catalog >> +endobj +2 0 obj +<< /Author () /CreationDate (D:20250704113420+00'00') /Creator (LaTeX with hyperref) /ModDate (D:20250704113420+00'00') /Producer (LuaTeX-1.16.0) >> +endobj +3 0 obj +<< /Subtype /XML /Type /Metadata /Length 808 >> +stream + + + + + + + + +endstream +endobj +4 0 obj +<< /Count 47 /Kids [ 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 10 0 R 11 0 R 12 0 R 13 0 R 14 0 R 15 0 R 16 0 R 17 0 R 18 0 R 19 0 R 20 0 R 21 0 R 22 0 R 23 0 R 24 0 R 25 0 R 26 0 R 27 0 R 28 0 R 29 0 R 30 0 R 31 0 R 32 0 R 33 0 R 34 0 R 35 0 R 36 0 R 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[<8b41d259c856cf2ef2f3fd39b8622ab8><8b41d259c856cf2ef2f3fd39b8622ab8>] >> +startxref +714700 +%%EOF diff --git a/uni/aak/Skript.txt b/uni/aak/Skript.txt new file mode 100644 index 0000000..371697a --- /dev/null +++ b/uni/aak/Skript.txt @@ -0,0 +1,2562 @@ +6 Einführung in die Komplexitätstheorie +In den vorherigen Kapiteln haben wir bereits einige theoretische Probleme kennenge- +lernt. Wir haben insbesondere gesehen, dass sich einige Probleme schneller lösen (bzw. +approximieren) lassen als andere. Außerdem scheint es Probleme zu geben, bei denen +man schnell den Mut verliert, eine exakte Lösung in Polynomialzeit zu berechnen. Diese +Probleme scheinen schwerer zu sein als andere. Was ist die Schwere eines Problems? Die +Komplexitätstheorie befasst sich noch weitergehend mit der Frage nach der Komplexität +von Problemen, dh. man versucht zu klassifizieren wie schwer Probleme sind. + In diesem Kapitel werden wir die Komplexitätsklassen P und N P einführen und wol- +len dann die Begriffe der N P -schwere und N P -vollständigkeit verstehen. Im Anschluss +werden wir einige Beispiele N P -vollständiger Probleme studieren. + + +6.1 Einleitung: Cliquenproblem +Als erstes Beispiel betrachten wir das sogenannte Cliquenproblem auf Graphen. Eine +Clique ist eine Teilmenge C ⊆ V , wobei {u, v} ∈ E für alle u, v ∈ C mit u 6= v. Das +Cliquenproblem können wir nun wie folgt angeben: + +Gegeben: Ein ungerichteter Graph G = (V, E) und eine Zahl k ≥ 1. + +Entscheide: Hat der Graph G eine Clique C ⊆ V mit mindestens k Knoten (d.h. mit + |C| ≥ k)? + +Dies ist ein sogenanntes Entscheidungsproblem, welches also als Antwort Ja oder Nein +gibt. + +Beispiel 6.1. Man betrachte den folgenden Graphen G = (V, E) mit V = {1, 2, 3, 4, 5} +und E = {{1, 2}, {1, 3}, {2, 3}, {3, 4}, {3, 5}}: + + 2 + b + + + + + 1 b b + 3 + + b b + 4 5 +Hier ist C = {1, 2, 3} eine Clique der Größe 3, aber C = {2, 3, 4} ist keine Clique. + + + +200 + 6.1 Einleitung: Cliquenproblem + + +Offensichtlich löst der folgende Algorithmus das Problem exakt: + +Algorithmus: + Teste für alle V 0 ⊆ V mit |V 0 | = k, ob V 0 eine Clique bildet. + +Wir bezeichnen mit T (G, k) seine Laufzeit für eine Eingabe (G, k). +Lemma 6.2. Der obige Algorithmus hat im schlimmsten Fall mindestens exponentielle +Laufzeit; d.h. T (G, k) ≥ 2|V |/2 für k = |V |/2. +Beweis. Für |V | = n, k = n/2 und n gerade: Der Test für jedes V 0 erfordert Ω(k 2 ) +Schritte. Es gibt nk viele k-elementige Teilmengen. Dann erhalten wir hier mit +   + n n(n − 1) · (n/2 + 1) + = ≥ 2n/2 + n/2 n/2(n/2 − 1) · 1 +mindestens eine exponentielle Laufzeit. + +Frage: Gibt es schnellere (bzw. insbesondere polynomielle) Algorithmen? +Definition 6.3. (i) Ein Alphabet Σ ist eine endliche Menge an Symbolen. Σ∗ ist die + Menge aller endlichen Wörter über Σ. + (ii) Für jedes x ∈ Σ∗ bezeichne |x| die Länge von x. +Bemerkung 6.4. Jeder Algorithmus arbeitet auf einem Alphabet (d.h. seine Eingaben +und Ausgaben sind Elemente aus Σ∗ ). Für einen Algorithmus A, bezeichnet A(x) die +Ausgabe zur Eingabe x ∈ Σ∗ . + Die Laufzeit eines Algorithmus A zur Eingabe x (gegeben durch die Anzahl der aus- +geführten Operationen) wird durch TA (x) bezeichnet. Die Worst Case Laufzeit zu Ein- +gaben der Größe n ist + TA (n) = max{TA (x)|x ∈ Σ∗ , |x| = n}. +Definition 6.5. (i) Eine Teilmenge L ⊆ Σ∗ heißt Sprache. + (ii) Ein Entscheidungsproblem ist ein Tripel (L, U, Σ) wobei L, U Sprachen sind und + L ⊆ U. +(iii) Ein Algorithmus A löst das Entscheidungsproblem (L, U, Σ) (oder auch A entschei- + det die Sprache L bezüglich U ), wenn für alle x ∈ U gilt: + – x ∈ L impliziert A(x) = 1, + – x 6∈ L impliziert A(x) = 0. + Häufig ist U = Σ∗ . Dann sagt man, dass A die Sprache L ⊆ Σ∗ entscheidet. Das +Cliquenproblem etwa können wir als Sprache (Menge von Wörtern) wie folgt darstellen: + CLIQU E = {u#v | u ∈ {0, 1}∗ stellt Adjazenzmatrix von G dar, + v ∈ {0, 1}∗ stellt k dar, + G hat Clique mit k Knoten }. + + + 201 + 6 Einführung in die Komplexitätstheorie + + +Hierbei ist Σ∗ = {0, 1}∗ die Menge aller endlichen Wörter (inklusive dem leeren Wort ) +über dem Alphabet Σ = {0, 1}. + +Bemerkung 6.6. u ist ein Wort der Länge O(|V |2 ), v ist ein Wort der Länge O(log |V |). + + +6.2 Komplexitätsklassen P, N P +Definition 6.7. Die Menge aller polynomiell entscheidbaren Sprachen bzw. Entschei- +dungsproblemen wird mit P bezeichnet: + + P = {L ⊆ Σ∗ | es existiert ein Algorithmus A, der L in Laufzeit + TA (n) = O(nd ) mit d = O(1) entscheidet}. + + Neben P gibt es unter anderem auch die Klasse N P . Diese wird häufig über nicht- +deterministische Algorithmen eingeführt (bei denen es zu jeder Eingabe eine endliche +Menge von Rechenwegen gibt). + +Definition 6.8. Ein nicht-deterministischer Algorithmus entscheidet eine Sprache L, + + • wenn es für jedes Wort w ∈ L mindestens einen Rechenweg gibt, der w akzeptiert, + + • wenn jeder akzeptierende Rechenweg in polynomieller Zeit läuft und + + • wenn für jedes Wort w 6∈ L jeder Rechenweg w ablehnt. + + Die Anzahl der Rechenwege kann dabei exponentiell von der Eingabe abhängen. + + Die Klasse N P ist nun genau die Menge aller Sprachen, die durch einen nicht-deterministischen +Algorithmus in Polynomialzeit entschieden werden können. Wir bedienen uns hier aller- +dings einer alternativen Definition der Klasse N P über polynomielle Verifizier. Die so +definierte Klassen von Problemen sind aber gleich. + +Definition 6.9. Es sei L ⊆ Σ∗ . Ein (deterministischer) Algorithmus, der auf Σ∗ × Σ∗ +arbeitet, heißt ein Verifizierer für L, wenn + + L = {x ∈ Σ∗ |∃c ∈ Σ∗ mit A(x, c) = 1}. + + Hierbei nennt man ein Wort c mit A(x, c) = 1 ein Zertifikat für x (bzw. für die +Aussage x ∈ L). + +Definition 6.10. Ein Algorithmus A ist ein polynomieller Verifizierer für eine Sprache +L, wenn eine Konstante d existiert, so dass es für jedes x ∈ L ein Zertifikat c gibt mit +TA (x, c) = O(|x|d ). + +Definition 6.11. Die Klasse aller polynomiell verifizierbaren Sprachen wird mit N P +bezeichnet; d.h. + + N P = {L ⊆ Σ∗ | es existiert ein polynomieller Verifizierer für L}. + + + +202 + 6.2 Komplexitätsklassen P, N P + + +Satz 6.12. Es gilt P ⊆ N P . + +Beweis. Zu einem Entscheidungsproblem L aus P gibt es einen Algorithmus A, der +L entscheidet. Diesen Algorithmus kann man als Verifizierer auffassen, der die zweite +Eingabe c einfach ignoriert. Damit ist L auch in N P . + + Nun können wir auf das Cliquenproblem zurückkommen und die folgende Aussage +beweisen. + +Lemma 6.13. Das Entscheidungsproblem k-Clique ist in N P . + +Beweis. Durch Angabe eines polynomiellen Verifizieres für k-Clique. Der Algorithmus +bekommt neben der Eingabe (einen Graph G = (V, E) und eine Zahl k) ein Zertifikat +(u.z. eine Teilmenge C ⊆ V ). Der Algorithmus testet dann, ob C eine Clique der Größe +≥ k ist. Der Test geht wie folgt: + + (1) prüfe für alle zweielementigen Teilmengen {u, v} ⊆ C, ob {u, v} ∈ E liegt. + + (2) gebe Ausgabe Ja aus, wenn jeder Test in (1) positiv ausfiel und |C| ≥ k ist. + Ansonsten Ausgabe Nein. + + Schritt (1) geht in Zeit O(|C|2 ) = O(|V |2 ) und die Kardinalitätsprüfung in (2) in +O(|C|) = O(|V |) Zeit. Da die Eingabelänge Ω(|V | + |E|) ist, ist die Laufzeit des Algo- +rithmus polynomiell. + + Nachdem wir nun gesehen haben, wie die Klassen P und N P definiert haben, können +wir den Begriff der Schwere genauer beleuchten. Dafür ist es notwendig zu verstehen, +wann ein Problem auf ein anderes Problem reduziert werden kann. Wir benötigen die +folgende Definition. + +Definition 6.14. Es seien L1 ⊆ Σ∗1 , L2 ⊆ Σ∗2 zwei Sprachen. + + (i) Eine Funktion f : Σ∗1 → Σ∗2 mit der Eigenschaft + + w ∈ L1 ⇔ f (w) ∈ L2 ∀w ∈ Σ∗1 + + heißt Transformation von L1 auf L2 . + + (ii) Eine solche Transformation heißt polynomielle Transformation, wenn es ein Po- + lynom p und einen Algorithmus A gibt, der für alle w ∈ Σ∗1 die Ausgabe f (w) der + Länge ≤ p(|w|) in (polynomieller) Zeit TA (w) ≤ p(|w|) berechnet. + +Dann heißt L1 polynomiell reduzierbar auf L2 . Als Notation verwenden wir L1 ≤ L2 . + +Endlich können wir den Begriff der N P -Schwere bzw. N P -Vollständigkeit formal fassen: + +Definition 6.15. (i) Eine Sprache L0 heißt N P -schwer, g.d.w. ∀L ∈ N P : L ≤ L0 . + + (ii) Eine Sprache L0 heißt N P -vollständig, g.d.w. L0 ∈ N P und ∀L ∈ N P gilt: L ≤ L0 . + + + + 203 + 6 Einführung in die Komplexitätstheorie + + +Eine fundamentale Eigenschaft der N P -vollständigkeit ist die folgende: + +Satz 6.16. Es sei L0 N P -vollständig. Dann gilt P = N P , g.d.w. L0 ∈ P . + +Beweis. ⇒: Es sei P = N P . Zeige dann, dass L0 ∈ P . + L0 ist NP-vollständig. Dann gilt insbesondere L0 ∈ N P . Wegen P = N P gilt dann +L0 ∈ P . + ⇐: Es sei L0 ∈ P . Zeige hier, dass P = N P . + Da L0 ∈ P ist, gibt es einen Algorithmus A0 , der L0 in Polynomzeit T0 entscheidet. +Es sei nun L ∈ N P beliebig gewählt. Zeige: L ∈ P . + Da L0 N P -vollständig ist und L ∈ N P , gilt L ≤ L0 . Sei also Af ein Algorithmus, +der die polynomielle Transformation f von L auf L0 in Polynomzeit Tf berechnet mit +u ∈ L ⇔ f (u) ∈ L0 . + Wir konstruieren A aus A0 und Af durch Hintereinanderschalten; siehe folgendes Bild: + + + Af A0 ja + u f (u) + nein + + Zeit Tf (|u|) Zeit T0 (|f (u)|) + + Dann akzeptiert A ein Wort u, g.d.w. A0 das Wort f (u) akzeptiert. Dies ist äquivalent +zu f (u) ∈ L0 bzw. u ∈ L. D.h. Algorithmus A entscheidet L mit Gesamtlaufzeit ≤ +Tf (|u|)+T0 (|f (u)|) ≤ T (|u|) für ein Polynom T . Hierbei verwende, dass |f (u)| ≤ poly(|u|) +und dass die Komposition von Polynomen wieder ein Polynom ergibt. + +Konsequenz von Satz 6.16: Wenn ein N P -vollständiges Entscheidungsproblem in P +liegt, dann sind alle N P Entscheidungsprobleme in P . + + Die gleiche Aussage gilt auch für ein N P -schweres Problem. +Offenes Problem der Komplexitätstheorie: + Gilt P = N P oder P 6= N P ? + + Dies ist eines der wichtigsten offenen Probleme der Informatik. Es wurde vom Clay +Mathematics Institute in die Liste der Millennium-Probleme aufgenommen; mit einem +Preisgeld von 1-Million Dollar: www.claymath.org. + +Bemerkung 6.17. Wenn L1 ≤ L2 und L2 ≤ L3 , dann gilt auch L1 ≤ L3 . + +Beweis. siehe Übung. + +Korollar 6.18. Ist L0 N P -vollständig, L0 ≤ L1 und L1 ∈ N P , so ist auch L1 N P - +vollständig. + + + +204 + 6.3 Definition von P und N P per Turingmaschine + + +Beweis. Zeige die folgende Behauptung: ∀L ∈ N P gilt L ≤ L1 . +Sei dazu L ∈ N P beliebig gewählt. Zeige dann die Aussage L ≤ L1 . Wegen der N P - +vollständigkeit von L0 und da L ∈ N P ist, gilt L ≤ L0 . Nach Voraussetzung gilt +zusätzlich L0 ≤ L1 . Da die Relation ≤ transitiv ist (siehe Bemerkung 6.17), gilt dann: +L ≤ L1 . + Da L1 ∈ N P wegen der Voraussetzung, ist daher L1 auch N P -vollständig. + + Das obige Ergebnis in Korollar 6.18 ermöglicht uns, N P -vollständige Probleme aus +bekannten N P -vollständigen Problemen zu konstruieren (bzw. deren N P -vollständigkeit +zu beweisen). Für diese Entscheidungsprobleme gibt es keine polynomiellen Algorithmen, +außer P = N P . + + +6.3 Definition von P und N P per Turingmaschine +Eine nicht-deterministische Turingmaschine (NDTM) hat die Form: + + M = (Q, Γ, q0 , δ, F ). + +Hierbei beschreibt Q die nicht-leere endliche Zustandsmenge, Γ das Arbeitsalphabet, +wobei Γ ⊃ Σ gilt (dabei ist Σ das Eingabealphabet). Die Zustandsmenge enthält den +Anfangszustand q0 ∈ Q und die Endzustandsmenge F ⊂ Q. Zuletzt beschreibt δ die +Übergangsfunktion: + δ : Q × Γ −→ 2Q×Γ×{−1,0,1} . +Es gilt, dass |δ(q, a)| = 0, 1, 2, . . . sein kann, man also mehrere mögliche Folgezustände +haben kann. + + Arbeitsfeld + + u a u + + + + + q + + + + Ein Übergang δ(q, a) → (q 0 , a0 , −1/1/0) besagt: im Zustand q mit a auf dem Arbeits- +feld (AF) drucke a0 auf AF, bewege den Schreiblesekopf (bzw. das AF) nach links, rechts +oder bleibe stehen und gehe danach in Zustand q 0 über. + Die möglichen Berechnungen einer NDTM wird über Konfigurationen (die die Band- +inschrift plus den Zustand enthalten) beschrieben, wobei eine Konfiguration c wie folgt +definiert ist: + + c ∈ Γ∗ × Q × Γ × Γ∗ + u q a v, + + + + 205 + 6 Einführung in die Komplexitätstheorie + + +wobei u und v Wörter sind, q ein Zustand und a der Buchstabe auf dem AF ist. + Für u = u0 b, v sei die Folgekonfiguration von uqav definiert durch: + + u0 q 0 ba0 v falls δ(q, a) = (a0 , 1, q 0 ) + u0 ba0 q 0 v falls δ(q, a) = (a0 , −1, q 0 ) + u0 bq 0 a0 v falls δ(q, a) = (a0 , 0, q 0 ) + +Als Notation verwenden wir C ` C 0 ; dabei ist C 0 die Folgekonfiguration von C. + +Berechnung zu Wörtern bzw. Eingaben: Hierbei ist die Startkonfiguration durch q0 [w +gegeben, wobei [ ein Sonderzeichen 'Blank' (bzw. ein Leerzeichen) darstellt mit [ ∈ Γ\Σ. +Eine Konfiguration C nennt man Stopkonfiguration, falls keine Folgekonfiguration von +C existiert. Eine Berechnung zu einer Eingabe w ∈ Σ∗ kann dann durch eine Folge +C0 , . . . , Ck von Konfigurationen dargestellt werden, wobei C0 die Startkonfiguration zu +einer Eingabe w ist, Ci ` Ci+1 , für alle i < k gilt und Ck eine Stopkonfiguration ist. +Eine Berechnung heißt genau dann akzeptierend, wenn der Zustand von Ck in F liegt. + Sei L ⊆ Σ∗ und T : N → N. M akzeptiert L [in nicht deterministischer Zeit T ] genau +dann wenn ∀w ∈ Σ∗ : (w ∈ L ⇐⇒ ∃ akzeptierende Berechnung von M zu w [mit Länge +≤ T (|w|)]. + Damit können wir die Zugehörigkeit zu der Komplexitätsklasse NP nun definieren. + +Definition 6.19. Eine Sprache L ⊂ Σ∗ ist in NP ⇐⇒ ∃ eine NDTM M über Alphabet +Γ ⊃ Σ und ein Polynom T , so dass gilt: M akzeptiert L in nicht deterministischer Zeit +T. + +Deterministische Turingmaschine (DTM) Falls |δ(q, a)| ≤ 1 ∀(q, a) ∈ Q × Γ, es also +höchstens einen gültigen Übergang aus jedem Zustand gibt, nennt man eine Turingma- +schine deterministisch. Die Berechnung der DTM ist analog zu NDTMs über Konfigu- +rationen definiert. + Eine DTM M entscheidet L [in deterministischer Zeit T ] genau dann wenn für alle +Wörter w ∈ Σ∗ die Berechnung von w mit einer Stopkonfiguration endet [und Länge +≤ T (|w|) hat]. Dabei liegt der Zustand der Stopkonfigurationen in F , genau dann wenn +w in L liegt. + Die Zugehörigkeit zu der Komplexitätsklasse P wird nun analog definiert. + +Definition 6.20. L ⊂ Σ∗ ∈ P ⇐⇒ ∃ eine DTM M über Alphabet Γ ⊃ Σ und ein Poly- +nom T , so dass M die Sprache L in Zeit T entscheidet. + + Nun bleibt zu zeigen, dass beide gezeigten Definitionen der Komplexitätsklasse N P +äquivalent sind, sie also die gleiche Klasse beschreiben. + +Satz 6.21. Die Klassen der Sprachen bezüglich Verifizierer und NDTM sind äquivalent +und beschreiben beide die gleiche Klasse N P . + +Beweis. ” ⊃ ” : Sei N eine NDTM, die eine Sprache L in Zeit nc für eine Konstante +c entscheidet. Für jedes x ∈ L gibt es eine Folge nicht-deterministischer Übergänge, + + + +206 + 6.4 Satisfiability Problem (SAT) + + +so dass N bei Eingabe x einen akzeptierenden Zustand qacc erreicht. Diese Folge kann +durch einen String u der Länge nc über einem Alphabet σ 0 mit O(1) vielen Buchstaben +repräsentiert werden (hierbei hängt die Kardinalität von σ 0 von der maximalen Anzahl +von Verzweigungen (d.h. |δ(q, a)|) ab. Dann folgt direkt, dass es eine DTM M gibt, die bei +Eingabe < x, u > genau dieselben Berechnungen durchführt, also die gleichen Übergänge +wie N macht. Für die Eingabe x ∈ L folgt also auch, dass M einen akzeptierenden +Zustand erreichen muss. Dann ist M also auch ein Verifizierer für L. + ” ⊂ ” : Sei L ⊆ Σ∗ gegeben und sei M ein Verifizierer für L. Sei p : N → N ein +Polynom, das die Länge der Zertifikate angibt. Sei nc die Laufzeit der DTM M , wobei +c wieder eine Konstante ist. Eine NDTM N , die w in Polynomzeit entscheidet, kann bei +Eingabe x ∈ Σ∗ wie folgt vorgehen: + Phase (1): Nutze nicht-deterministische Entscheidungen um einen beliebigen String u +der Länge poly(|x|), also polynomiell in |x|, auf ein Arbeitsband zu schreiben. + Phase (2): Simuliere M bei Eingabe < x, u >. Dann folgt direkt, dass N in Polynomzeit +entscheidet, da M diese Eingabe in Polynomzeit verifizieren kann. + + +6.4 Satisfiability Problem (SAT) +Der Kanadier Stephen A. Cook begründete 1971 die Klasse der N P -vollständigen Proble- +me in N P , indem er zeigte, dass ein N P -vollständiges Problem existiert. Dieses Problem +ist das sogenannte Satisfiability Problem (SAT). Leonid Levin hatte einen vergleichbaren +Satz 1973 veröffentlicht. + +Satisfiability Problem (SAT) + + Das Satisfiability Problem (SAT) formalisiert die Frage nach der Erfüllbarkeit eines +aussagenlogischen Ausdrucks. Aussagenlogische Ausdrücke sind zusammengesetzt aus +Variablen und den elementaren Operatoren/Junktoren ∧ (und), ∨ (oder) und ¬ (nicht). +Die Variablen sind Boolesche Variablen und wir notieren sie als xi, dh. x0, x1, x10, x11, . . . +und so weiter. Für jede Variable xi existieren zwei Literale. Diese sind das positive Lite- +ral xi und das negative Literal ¬xi. Aus den Literalen setzt man mit dem elementaren +Operator ∨ (oder) sogenannte Klauseln zusammen, dh. disjunktive Ausdrücke der Form +(y1 ∨ · · · ∨ yk ) mit Literalen y1 , . . . , yk . + Ein Ausdruck α = C1 ∧ . . . ∧ Cm mit Klauseln C1 , . . . , Cm heißt Ausdruck in Kon- +junktiver Normalform (KNF). Ein Ausdruck α heißt erfüllbar, g.d.w. eine Belegung φ +der Variablen in α mit true, f alse existiert, so dass α unter der Belegung φ zu true +ausgewertet wird (siehe Beispiel 6.23). + Nun können wir das SAT-Problem als Sprache wie folgt angeben: + + SAT = {α | α Boolescher Ausdruck in KNF, α erfüllbar } +Bemerkung 6.22. SAT ⊂ {x, 1, 0, ∨, ∧, ¬, (, )}∗ . +Beispiel 6.23. Betrachte α = (x1∨x10)∧(x1∨¬x10)∧(x10). Wir erhalten die folgende +Wahrheitstabelle: + + + + 207 + 6 Einführung in die Komplexitätstheorie + + + x1 x10 α + false false false + true false false + false true false + true true true + +Die Belegung ψ(x1) = ψ(x10) = true ist eine erfüllende Belegung. + +Satz 6.24. (Cook, 1971/Levin, 1973): SAT ist N P -vollständig. + +Beweis. SAT ∈ NP Wir skizzieren dies für eine entsprechende nichtdeterministische Tu- + ringmaschine mit den Zwischenergebnissen auf dem Turingband in Abb. 6.1. Salopp + gesagt raten“ wir nichtdeterministisch eine Belegung der Variablen und prüfen + ” + dann, ob sie den Ausdruck erfüllt. + + α + (1) ↓ det. + α#Liste der auftretenden Variablen + (2) ↓ nichtdet. + α#Belegung der Variablen durch wahr/falsch: + x101 → T 101 oder F 101. + (3) ↓ det. + α#Wert von α + (4) ↓ det. + F , wenn α = wahr; dummy-Zustand, falls α = falsch. + + Abb. 6.1: SAT ∈ NP: Alle vier Schritte sind in polynomieller Zeit ausführbar. + + +L ∈ NP =⇒ L ≤ SAT Sei M eine NDTM und akzeptiere L in nichtdeterministischer + Zeit T (für ein geeignetes Polynom T ). Wir suchen eine Transformation u → αu , + die polynomzeitberechenbar ist mit u ∈ L ⇐⇒ αu ∈ SAT. + Idee: αu wird so beschrieben, dass M für u in genau T (u) Schritten eine akzeptie- + rende Stoppkonfiguration erreicht. Sei also + + M = (Q, Γ, q0 , δ, F ), + + wobei Q = {q0 , . . . , qs }, F = {qr , . . . , qs }, Γ = {a0 , . . . , am }, a0 = [. Ohne Ein- + schränkung der Allgemeinheit gelte: Akzeptiert M das Wort u = aj1 . . . aj|u| mit + einer Berechnung der Länge höchstens T (|u|), so akzeptiere M das Wort mit einer + Berechung der Länge genau gleich T (|u|). + Wir stellen δ durch eine Folge von Zeilen (Fünftupeln) z1 , . . . , z% der Form qaq 0 a0 β + dar (für (q 0 , a0 , β) ∈ δ(q, a)). O.B.d.A betrachten wir nur Turingmaschinen, die + nicht links vom Feld 0 arbeiten. Der zu konstruierende Ausdruck in KNF α enthält + Variablen, deren Bedeutung in Tabelle 6.1 zusammengefasst ist. + + + +208 + 6.4 Satisfiability Problem (SAT) + + + Variablen in αk Bedeutung (wahr, falls. . . ) + ztk nach t Schritten wird Zustand qk erreicht + sti nach t Schritten ist Feld Nr. i das Arbeitsfeld + btl nach t Schritten wird Zeile zl ausgeführt + atij nach t Schritten steht auf Feld Nr. i der Buchstabe aj , +Hierbei ist 0 ≤ t, i ≤ T (|u|), 0 ≤ k ≤ S, 1 ≤ l ≤ %, 0 ≤ j ≤ m. + Tab. 6.1: Variablen in αk + + + + Die Anzahl der Variablen ist nach oben durch c · T (|u|)2 beschränkt, wobei c eine + von der Turingmaschine abhängige Konstante ist. + Wir bauen nun einen Ausdruck αu auf mit + + αu = αAnfang ∧ αEnde ∧ αEindeutig ∧ αÜbergang . + + Das Ziel ist, dass erfüllende Belegungen von αu den akzeptierenden Berechnungen + in Zeit T (|u|) von M für u = aj1 . . . aj|u| entsprechen sollen. Wir setzen + + αAnfang = z00 ∧ s00 ∧ a000 ∧ a01j1 ∧ a02j2 ∧ · · · ∧ a0|u|j|u| ∧ a0(|u|+1)0 ∧ · · · ∧ a0 T (|u|) 0 + αEnde = zT (|u|) r ∨ · · · ∨ zT (|u|) s + genau ein Zustand“ + ^ + αEindeutig = + ” + 0≤t≤T (|u|) + + ∧ genau ein Arbeitsfeld“ + ” + ∧ für alle Felder i = 0, . . . , T (|u|) genau ein Buchstabe“ +  + ” + Den ersten Teil, genau ein Zustand“, schreiben wir als + ” + ^ + (zt0 ∨ · · · ∨ zts ) ∧ ¬(zti ∧ ztj ), + | {z } + i6=j + (¬zti ∨¬ztj ) + + die anderen Bedingungen analog. Wir setzen + ^ + αÜbergang = αt , + 0≤t≤T (|u|) + + wobei αt Folgendes modelliert: + • Buchstaben auf nicht-Arbeitsfeldern bleiben unverändert + • Falls das Arbeitsfeld Nr. i hat und die l. Zeile qkl ajl qk̃l aj̃l βl ausgeführt wird, + gilt: + a) nach t Schritten wird qkl erreicht + b) nach t Schritten steht auf Feld i der Buchstabe aj l + c) nach t + 1 Schritten wird qk̃l erreicht + + + + 209 + 6 Einführung in die Komplexitätstheorie + + + d) nach t + 1 Schritten steht auf Feld i Buchstabe aj̃l + e) nach t + 1 Schritten hat das Arbeitsfeld die Nummer i + βl . + Wir erhalten insgesamt: + ^ ^  + αt = ((¬sti ) ∧ atij ) → a(t+1)ij + 0≤i≤T (|u|) 0≤j≤m + ^  + ∧ (sti ∧ btl ) → (ztkl ∧ atijl ∧ z(t+1)k̃l ∧ a(t+1)i̃l ∧ s(t+1)(i+βl ) ) + 1 0 : ∃x0 ≥ 0 : ∀x ≥ xo : |f (x)| ≤ c · |g(x)| + +Man kann dies auch als fg(x) + (x) + → 0 für x → ∞. Nach dieser Definition gilt also auch +dass δn = o(n), weswegen die beiden Schreibweisen austauschbar verwendet werden. Die +Originale Formulierung der ETH verwendet die δn Notation, weswegen sie in unserer +Formulierung der Theoreme verwendet wurde. + + +6.14 Hamiltonkreis Problem +Das Hamitonkreis Problem (HK) is wie folgt definiert: + +Gegeben: Ein ungerichteter Graph G = (V, E) mit V = {v1 , . . . , vn }. +Entscheide: Gibt es einen Hamiltonschen Kreis; d.h. eine Permutation π der Knoten + (vπ(1) , . . . , vπ(n) ) mit {vπ(i) , vπ(i mod n+1) } ∈ E für alle i = 1, . . . , n? + +Beispiel 6.41. Der Graph in dem folgenden Bild hat einen entsprechenden Hamilton- +kreis; dieser ist fettgezeichnet. + + + +220 + 6.14 Hamiltonkreis Problem + + + + +Satz 6.42. Das Hamiltonkreis Problem ist NP-vollständig. + +Bemerkung 6.43. Die gleiche Aussage gilt auch für gerichtete Graphen. + +Beweis. (a) HK is in NP: Wähle als Zertifikat eine Permutation π der Knoten und teste, +ob {vpi(i) , vπ(i+1 mod n) } ∈ E ist für alle i = 1, . . . , n. + (b) 3−SAT 0 ≤ HK: Gegeben seine SAT Formel mit genau 3 Literalen pro Klausel (ist +ebenfalls NP-vollständig). Zur Beweisidee geben wir die folgende Graph-Konstruktionen +an. Als erstes verwende die folgende A-Komponente: + + 1 3 + + + + + 5 6 7 8 + + + + + 2 4 + + + + + Dann gibt es zwei Möglichkeiten, die Komponente A zu durchlaufen; siehe die folgen- +den zwei Abbildungen: + + + 1 3 + + + + + 2 4 + + + + + 1 3 + + + + + 2 4 + + + + + Für jede Klausel verwenden wir die folgende Komponente B: + + + + 221 + 6 Einführung in die Komplexitätstheorie + + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + Eine der vielen Möglichkeiten, wie B durchlaufen werden kann, ist im folgenden Bild +dargestellt. + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + Insgesamt gibt es für jede nicht-leere Teilmenge der rechten Kanten {u1 , u2 }, {u2 , u3 }, {u3 , u4 } +eine Möglichkeit die Komponente B zu durchlaufen. Bei einer leeren Teilmenge dieser +Kanten funktioniert dies aber nicht; dies entsprecht dem Fall, dass alle Literale falsch +gewählt worden sind. Als Kurzform für die Spezialkomponenten A und B verwenden +wir: + + 1 3 + + + 5 + + + 2 4 + + + + +222 + 6.14 Hamiltonkreis Problem + + 1 + + + + + 2 + + + 5 + + + 3 + + + + + 4 + + + + Im folgenden Beispiel zeigen wir die Konstruktion des Graphen G zu einem Booleschen +Ausdruck F . + +Beispiel 6.44. Ausdruck F = (x1 ∨ ¬x2 ∨ x3 ) ∧ (¬x1 ∨ x2 ∨ ¬x3 ) ∧ (¬x1 ∨ ¬x2 ∨ x3 ) und +der zugehörige konstruierte Graph G: + + + 13 + 1 + 10 + 2 + 14 + 3 4 + 15 + + 11 5 + 6 16 + + 7 17 + + 8 + 18 + 12 9 + + + + + Für die Variablensetzung von Variable xi wählt man im rechten Pfad im obigen Bild +jeweils zwischen zwei Knoten vi , wi für i = 1, . . . , n die rechte oder linke Kopie der +Kante {vi , wi }. Die A-Komponenten werden nun genutzt um die Verbindung zwischen +den Variablen und den Literalen in den B-Komponenten zu den entsprechenden Klauseln +herzustellen. Dazu bezeichnen wir die u Knoten in der B-Komponente zu Klausel Fi mit +ui1 , ui2 , ui3 und ui4 . + Wenn das j.te Literal in Klausel Fi gleich xk ist, dann verbinden wir die Kante +{uij , uij+1 } via einer A-Komponente mit der linken Kopie von {vk , wk }. Wenn das j.te +Literal gleich ¬xk , so verbinden wir die Kante {uij , uij+1 } mit der rechten Kopie. + Formal beweist man Ende die folgende Aussage: F is erfüllbar genau dann, wenn G +einen Hamiltonkreis besitzt. + + Die Details findet man im Buch von Papadimitriou und Steiglitz. + + + + 223 + 6 Einführung in die Komplexitätstheorie + + +Übungsaufgaben +Übung 6.45. Die Eingabe des Problems SAT3 ist ein boolscher Ausdruck in KNF, für +den entschieden werden soll, ob es mindestens 3 erfüllende Belegungen für diese Formel +gibt. + Zeigen Sie, dass SAT3 N P -vollständig ist. +Übung 6.46. Eine NAE-k-SAT (Not-All-Equal) Formel hat folgende Form m i i + V + i=1 (z1 , . . . , zk ), +wobei eine Klausel (z1i , . . . , zki ) mit Literalen z1i , . . . , zki genau dann erfüllt ist, wenn min- +destens ein Literal zu wahr und mindestens ein Literal zu falsch ausgewertet wird. + Beispiel: Für die NAE-3-SAT Formel (x, y, z) ∧ (¬x, y, z) ist durch x = y = z = true +keine erfüllende Belegung gegeben, da in diesem Fall die erste Klausel nicht erfüllt ist. +Bei der Belegung x = y = true und z = false hingegen ist die Formel erfüllt. + Zeigen Sie in einem ersten Schritt, dass NAE-4-SAT NP-vollständig ist, und in einem +zweiten Reduktionsschritt zeigen, dass ebenfalls NAE-3-SAT NP-vollständig ist. +Übung 6.47. Ein Independent Set in einem Graphen G = (V, E) ist eine Menge von +Knoten I, so dass für je zwei Knoten i, j ∈ I gilt, dass {i, j} 6∈ E. Für das Problem +Independent Set ist ein Graph G = (V, E) sowie eine Zahl k ∈ N gegeben und es soll +entschieden werden, ob es in G ein Independent Set mit k Knoten gibt. + Zeigen Sie, dass das Problem Independent Set N P -vollständig ist. +Übung 6.48. Ein Vertex Cover in einem Graphen G = (V, E) ist eine Menge von +Knoten C ⊆ V so dass für jede Kante {v, w} ∈ E gilt das v ∈ C oder w ∈ C. Für das +Problem Vertex Cover ist ein Graph G = (V, E) sowie eine Zahl k ∈ N gegeben und +es soll entschieden werden, ob es in G ein Vertex Cover mit k Knoten gibt. + Zeigen Sie, dass das Problem Vertex Cover N P -vollständig ist. +Übung 6.49. In dem Problem Clique-Member ist ein Graph G = (V, E), ein Knoten +v ∈ V sowie eine Zahl k ∈ N gegeben und es soll entschieden werden, ob es eine Clique +mit k Knoten gibt, die den Knoten v enthält. + Zeigen Sie, dass Clique-Member N P -vollständig ist. +Übung 6.50. In dem Problem Clique-Nomember ist ein Graph G = (V, E), ein +Knoten v ∈ V sowie eine Zahl k ∈ N gegeben und es soll entschieden werden, ob es eine +Clique mit k Knoten gibt, die den Knoten v nicht enthält. + Zeigen Sie, dass Clique-Nomember N P -vollständig ist, indem Sie eine Reduktion +von Clique auf Clique-Nomember angeben. +Übung 6.51. In dem Problem Z-Clique ist ein zusammenhängender Graph G = +(V, E) und eine Zahl k ∈ N gegeben und es soll entschieden werden, ob G eine Clique +mit k Knoten enthält. + Zeigen Sie, dass Z-Clique N P -vollständig ist. +Übung 6.52. Zeigen Sie, dass das Partition Problem ebenfalls NP-schwer ist, wenn +eine Teilmenge A00 ⊆ A gesucht ist mit + X X + a=3 a. + a∈A00 a∈A\A00 + + + + +224 + 6.14 Hamiltonkreis Problem + + +Übung 6.53. Für das Problem Subset Sum Cardinality seien n ganze Zahlen +c1 , . . . , cn ∈ N>0 (wobei n gerade sei) sowie eine Zahl K ∈ N gegeben. Es + P soll entschieden +werden, ob es eine Teilmenge S ⊂ {1, . . . , n} gibt mit |S| = n/2 und i∈S ci = K. + Zeigen Sie, dass das Problem Subset Sum Cardinality N P -vollständig ist. + +Übung 6.54. Eine Eingabe des Problems Even-Knapsack ist eine Menge von Items +I, sowie ein Rucksack mit gerader Größe B (d.h es gibt b ∈ N mit B = 2b), sowie +ein Profitwert P ∈ N. Jedes Item hat eine Größe si ∈ N und einen P Profit pi ∈ N. +Das Entscheidungsproblem ist eine Teilmenge I 0 ⊆ I zu finden mit + i∈I 0 si ≤ B und + i∈I 0 pi ≥ P . +P + Zeigen Sie, dass Even-Knapsack N P -vollständig ist. + +Übung 6.55. Die Probleme Multiple-Choice-Knapsack (kurz MC-Knapsack) ist wie folgt +definiert: Gegeben sind endliche Mengen C1 , . . . Ck ⊆ N × N von Items, eine Kapazität +B ∈ N und ein Zielprofit P ∈ N. Wir bezeichnen Ci als Klasse von Items und für ein +Item (w, p) ∈ Ci ist w das Gewicht und p der Profit des Items. Es soll entschieden +werden, ob aus jeder Klasse Ci genau ein Item (wi , pi ) gewählt werden kann, sodass die +gewählten Items gemeinsam + P den Zielprofit erfüllen ohne die Kapazität zu überschreiten, +d.h. i∈[k] pi ≥ P und i∈[k] wi ≤ B. + P + Das Problem ist in N P . Zeigen Sie: MC-Knapsack ist N P -vollständig + + + + + 225 + 7 Approximative Algorithmen +Das letzte Kapitel hat uns Einblicke in die Schwierigkeit der polynomiellen Berechen- +barkeit gegeben. Wir wissen nun insbesondere, dass es viele Probleme gibt, deren exakte +Lösung eine große, nicht-polynomielle Laufzeit benötigt - es sei denn, es gilt P = N P . +In Ermangelung einer Eingebung zu der Frage, ob P = N P oder P 6= N P gilt, hat sich +die Disziplin der Approximativen Algorithmen entwickelt; anstatt eine exakte Lösung zu +verlangen, lässt man die Berechnung geringfügig schlechterer Lösungen zu, welche sich +aber deutlich schneller berechnen lassen. Die exakte Lösung wird in diesem Sinne nur +noch approximiert. + In diesem Kapitel wollen wir einige solche Approximationsalgorithmen studieren. + + +7.1 Traveling Salesman Problem +Gegeben: Eine Menge V = {1, . . . , n}, Distanzen d(i, j) ∈ Z+ ∪ {∞} für alle i, j ∈ + {1, . . . , n} + +Gesucht: Eine minimale Rundreise, die jeden Knoten genau einmal besucht. + +Formal: Eine Lösung P + ist gegeben durch eine Permutation Π von {1, . . . , n}, wobei die + Länge der Tour ni=1 d(Π(i), Π(i + 1)) mit Π(n + 1) = Π(1) ist. + +Beispiel 7.1. Betrachte den folgenden Graphen mit 4 Knoten: + + 1 b 1 b 2 + 2 + 1 1 + 2 + b b + + 4 1 3 + Eine optimale Rundreise ist [1, 2, 3, 4, 1] mit Länge 4. + +Satz 7.2. Das Entscheidungsproblem zum Traveling Salesman Problem, ob eine Rund- +reise mit Länge ≤ L existiert, ist NP-vollständig. + +Beweis. Dies folgt durch eine Reduktion vom Hamiltonkreis Problem: Wähle für jede +Kante {u, v} ∈ E den Wert d(u, v) = 1 und für alle anderen Paare {u, v} 6∈ E den Wert +d(u, v) = |V | + 1 sowie L = |V |. + + + +226 + 7.1 Traveling Salesman Problem + + + Durch Modifikation der Distanzen d(u, v) = ∞ für alle {u, v} 6∈ E, kann man sogar +zeigen, dass es keinen approximativen Algorithmus für das allgemeine TSP Problem mit +beschränkter Approximationsgüte gibt. + Deswegen betrachten wir den symmetrischer Fall des TSP mit d(i, j) = d(j, i) und +mit Erfüllung der Dreiecksungleichung (kurz: ∆-Ungleichung) d(i, j) ≤ d(i, k) + d(k, j); +siehe auch folgendes Bild: + k + b + + + + + i b b + j +Bemerkung 7.3. Das TSP bleibt NP-vollständig im symmetrischen Fall mit Dreiecks- +ungleichung. +Definition 7.4. Es sei G ein Multigraph mit parallelen Kanten. Ein Eulerscher Kreis +ist ein Kreis, der jede Kante von G genau einmal besucht. +Beispiel 7.5. (Königsberger Brückenproblem) + + A + + B C + + D +Frage: Gibt es einen Rundweg, bei dem man über jede der 7 Brücken genau einmal +läuft? + Wir können das Brückenproblem in ein Graphproblem wie folgt transformieren: + + + A + + B C + + D + + 227 + 7 Approximative Algorithmen + + + Frage: Gibt es einen Eulerschen Kreis in dem Graphen? + Die Frage nach einem Eulerschen Kreis lässt sich durch das folgende Kriterium beant- +worten. + +Satz 7.6. Es sei G ein zusammenhängender Multigraph. Dann sind äquivalent: + + (a) G ist Eulersch (d.h. besitzt einen Eulerschen Kreis). + + (b) Jeder Knoten in G hat geraden Grad. + + (c) Die Kantenmenge von G kann in disjunkte Kreise zerlegt werden. + +Beweis. zur Übung. + + Mit Hilfe von dem obigen Kriterium sieht man nun, dass der obige Graph zum +Königsberger Brückenproblem Knoten mit ungeradem Grad besitzt und es daher keinen +entsprechenden Rundweg gibt. + Wir betrachten nun den folgenden Algorithmus: + +Algorithmus ∆T SP1 (V = {1,...,n}, D = (d(i,j)) ) + 1 Berechne einen minimalen spannenden Baum T des Graphen + 2 Kn mit Distanzen d(i, j); + 3 Konstruiere einen Multigraphen G aus T durch Verdoppeln + 4 aller Kanten; + 5 Bestimme einen Eulerschen Kreis K in G; + 6 Bestimme eine Rundreise R aus K durch Abkürzungen; + 7 return Rundreise R. + +Beispiel 7.7. Betrachte den Graphen mit V = {A, B, C, D, E}; siehe folgendes Bild +(links). + + B + b + B + b + + + 1 2 1 + + C b 1 b A C b 1 b A + 2 3 + + 2 3 3 2 2 2 + + b b b b + + D 2 E D E + + Eine optimale Tour ist [C, B, D, E, A, C] der Länge 8. Der Algorithmus berechnet einen +Baum T mit Gewicht w(T ) = 6; siehe obiges Bild (rechts). Wir erhalten den Graphen +G durch Verdopplung der Kanten mit w(G) = 12; siehe folgendes Bild: + + + +228 + 7.1 Traveling Salesman Problem + + B + b + B + b + + 1 1 + 1 1 + C b 1 b A C b 1 b A + 1 1 + 2 2 2 3 2 + 2 2 2 + 2 2 + + b b b b + + D E D E + + Als Eulerschen Kreis erhalten wir K = [C, A, E, A, C, B, C, D, C] mit Länge d(K) = +12; und durch die Abkürzungen (d.h. Überspringen der schon vorher besuchten Knoten) +erhalten wir die Rundreise R = [C, A, E, B, D, C] mit Länge d(R) = 10. + +Satz 7.8. Der Algorithmus ∆T SP1 hat eine multiplikative Güte 2; d.h. d(R) ≤ 2OP T (I), +wobei OP T (I) die Länge einer minimalen Rundreise zu der Eingabe I mit V = {1, . . . , n} +und Distanzen D = (d(i, j)) ist. + +Beweis. Die Behauptung folgt aus den folgenden Ungleichungen: + + (1) Das Gewicht w(T ) ≤ OP T (I), da eine optimale Rundreise nach Weglassen ei- + ner Kante einen spannenden Baum ergibt und w(T ) das kleinste Gewicht eines + spannenden Baums ist. + + (2) d(K) = 2w(T ) ≤ 2OP T (I). Die erste Gleichung folgt wegen der Verdopplung der + Kanten und die Ungleichung wegen (1). + + (3) d(R) ≤ d(K). Das gilt, da wegen der ∆-Ungleichung die folgende Ungleichung gilt: + d(i, j) ≤ d(i, `1 ) + d(`1 , `2 ) + . . . + d(`t , j); siehe auch folgendes Bild: + + `1 + b + `t + b + + + + + b b + + i j + +Formal zeigt man die obige Ungleichung per Induktion nach t ≥ 1. + +Bemerkung 7.9. Es gibt Beispiele für Eingaben I zu dem Traveling Salesman Problem +mit OP T (I) = n, bei denen der Algorithmus ∆T SP1 eine Rundreise der Länge 2n − 2 +berechnet. + +Beweis. Zur Beweisidee geben wir hier ein Beispiel mit n = 8 Knoten. Betrachte ei- +nen Stern mit Knoten 8 in der Mitte und Knoten 1, . . . , 7 außen, die über einen Kreis +miteinander verbunden sind. Formal haben wir V = {1, . . . , 8} und E = {{i, 8}|i = +1, . . . , 7} ∪ {{i, i + 1}|i = 1, . . . , 6} ∪ {{7, 1}}. Alle Kanten haben die Distanz 1; die +anderen Nicht-Kanten haben Distanz 2. + + + + 229 + 7 Approximative Algorithmen + + + Eine optimale TSP Tour hat die Form [1, 2, 3, 4, 5, 6, 7, 8, 1] der Länge 8. Dagegen +kann der MST Algorithmus einen Stern mit 8 als Mittelpunkt und allen anderen Knoten +außen generieren; d.h. T = (V, E 0 ) hat die Kantenmenge E 0 = {{i, 8}|i = 1, . . . , 7}. +Durch Verdopplung der Kanten erhalten wir einen Multigraphen mit 2n − 2 Kanten. +Eine mögliche Eulertour ist + + [8, 1, 8, 3, 8, 5, 8, 7, 8, 2, 8, 4, 8, 6, 8]. + +Der Algorithmus erzeugt dann über die Abkürzungen die folgende Tour: + + [8, 1, 3, 5, 7, 2, 4, 6, 8] + +mit Länge 14 = 2n − 2. + +Definition 7.10. Ein Matching in einem ungerichteten Graphen G = (V, E) ist ei- +ne Teilmenge E 0 ⊆ E von Kanten, von denen keine zwei Kanten einen gemeinsa- +men +P Endpunkt gemeinsam haben. 0Bei Gewichten d(i, j) auf den Kanten sei d(E ) = + 0 + + {i,j}∈E 0 d(i, j) das Gewicht von E . Ein Matching ist perfekt, falls jeder Knoten v ∈ V +dabei überdeckt wird; d.h. 2|E 0 | = |V |. + +Beispiel 7.11. Ein bipartiter Graph G und ein perfektes Matching: + b b + + + + + b b + + + + + b b + + + + +Satz 7.12. (Lawler 1976): Ein perfektes Matching in einem Graphen G = (V, E) mit +minimalem Gewicht kann in Zeit O(|V |3 ) berechnet werden. + +Beweis. siehe z.B. Vorlesung Discrete Optimization. + + Eine bessere Approximationsgüte erhalten wir, in dem wir die Knoten mit ungeraden +Grad optimaler durch Kanten augmentieren. Hier hilft uns ein Matching mit minimalem +Gewicht. Man beachte, dass die Anzahl der Knoten mit ungeradem Grad gerade ist; +siehe auch Lemma 5.4. + +Algorithmus ∆T SP2 (V = {1,...,n}, D = (d(i,j)) ) + 1 Berechne einen minimalen spannenden Baum T des + 2 Graphen Kn mit Gewichten d(i, j); + 3 Bestimme die Menge X der Knoten in T, + 4 die ungeraden Grad haben; + 5 Bilde den vollständigen Graphen H auf X + + + +230 + 7.1 Traveling Salesman Problem + + + 6 mit Gewichten d(i, j) für i, j ∈ X mit i 6= j. + 7 Bestimme ein perfektes Matching K in H mit minimalem + 8 Gewicht. + 9 Bilde den Multigraphen G, der aus T durch +10 Hinzufügen aller Kanten aus K entsteht. +11 Bestimme einen Eulerschen Kreis C in G; +12 Bestimme eine Rundreise R aus C durch Abkürzungen; +13 return Rundreise R. + +Satz 7.13. (Christofides): Der Algorithmus ∆T SP2 hat eine multiplikative Güte 1.5; +d.h. d(R) ≤ 1.5OP T (I). + +Beweis. Es gilt d(C) = w(T ) + d(K) ≤ OP T (I) + d(K). Man zeige nun, dass d(K) ≤ +OP T (I)/2. Daraus folgt dann die Behauptung. + Sei dazu (i1 , . . . , i2m ) mit |X| = 2m die Reihenfolge, in der die Knoten von X in einer +optimalen Tour δ von I durchlaufen werden. Betrachte die Matchings: + + M1 = {{i1 , i2 }, {i3 , i4 }, . . . , {i2m−1 , i2m }} + M2 = {{i2 , i3 }, {i4 , i5 }, . . . , {i2m , i1 }} + + Betrachte eine optimale Tour und die Knoten aus X: + + x + b b + y + i1 b b + i2 + b b + + + + + b b + + + + + i6 b b i3 + b + + + b + + b + + + b b + + i5 i4 + b + + + + + Wegen der ∆-Ungleichung gilt: OP T (I) ≥ d(i1 , i2 ) + d(i2 , i3 ) + . . . + d(i2m , i1 ) = +d(M1 ) + d(M2 ). Daher gilt: OP T (I) ≥ d(M1 ) + d(M2 ) ≥ 2d(K), da K ein perfektes +Matching bezüglich X mit minimalem Gewicht ist. Deswegen gilt: d(K) ≤ OP T (I)/2 +und damit die Behauptung. + +Beispiel 7.14. Zu unserem Graphen im obigen Beispiel erhalten wir den folgenden +induzierten Graphen H rechts mit einem perfektem Matching K und Gewicht w(K) = 3: + + + + 231 + 7 Approximative Algorithmen + + Bb + + + 1 2 H + B b + 3 b + E + C b 1 b A + 2 3 + 2 3 + 1 2 + 2 3 3 2 K + MST b b + + C 2 D + b b + + D 2 E + + Als Multigraphen G erhalten wir dann + B + b + B + b + + 1 1 2 + 1 1 + C b + 1 b A C b + 1 b A + + + 2 2 2 2 + + + b + 2 b b + 2 b + + D E D E + + Ein Eulerkreis in G ist C = [C, B, C, A, E, D, C] und nach den Abkürzungen erhalten +wir R = [C, B, A, E, D, C] mit Länge d(R) = 9. +Lemma 7.15. Es gibt Beispiele für Eingaben I zu dem Traveling Salesman Problem mit +OP T (I) = n, bei denen der Algorithmus ∆T SP2 von Christofides eine Rundreise der +Länge (n − 1) + bn/2c berechnet. +Beweis. zur Übung. + + +7.2 Rucksackproblem +Dass das Rucksackproblem N P -vollständig ist, haben wir schon am Ende des letzten Ka- +pitels gesehen. Hier wollen wir nun einige einfache Approximationsalgorithmen studieren. + +Gegeben: + • n Gegenstände mit Gewichten w0 , . . . , wn−1 ∈ N und Gewinnen p0 , . . . , pn−1 ∈ N. + + • Rucksack der Kapazität B ∈ N. +Gesucht: Teilmenge I ⊆ {0, . . . , n − 1} mit i∈I wi ≤ B und + P + + X X X + pi = max{ pi |I 0 ⊆ {0, . . . , n − 1}, wi ≤ B}. + i∈I i∈I 0 i∈I 0 + + + + +232 + 7.2 Rucksackproblem + + +Satz 7.16. Das Entscheidungsproblem zum Rucksackproblem ist NP-vollständig; sogar +wenn wi = pi für alle i = 0, . . . , n − 1. + +Beweis. siehe Kapitel Einführung in die Komplexitätstheorie. + + Zunächst betrachten wir einen einfachen Greedy Algorithmus (GA): + +Algorithmus GA(I = ((w0 , p0 ), . . . , (wn−1 , pn−1 ), B)) + 1 sortiere die Gegenstände, so dass p0 /w0 ≥ p1 /w1 ≥ . . . ≥ pn−1 /wn−1 ; + 2 setze S = ∅; + 3 for i = + P0 to n − 1 do + 4 if ( j∈S wj ) + wi ≤ B then + 5 S = S ∪ {i} + 6 fi + 7 od + 8 return(S). + + Es sei GA(I) der Gesamtgewinn der durch Algorithmus GA berechneten Lösung S ⊆ +{0, . . . , n − 1} und OP T (I) der Gesamtgewinn einer optimalen Lösung zur Eingabe I. + +Satz 7.17. (a) Die Laufzeit des Algorithmus GA ist O(n log n). + + (b) Der Algorithmus GA hat eine multiplikative Güte größer gleich B−1 (d.h. OP T (I)/GA(I) ≥ + (B − 1)). + Der Algorithmus GA hat daher keine konstante Worst-Case Güte. + +Beweis. Zu (a): hierzu speichere bei jedem Hinzufügen eines Gegenstands die momentan +benutze Kapazität K. Dann geht Schritt (2 − 8) in O(n) Zeit. Die Gesamtlaufzeit ergibt +sich durch das Sortieren in O(n log n). + Zu (b): Betrachte das folgende Beispiel: I = ((w0 , p0 ) = (1, 1), (w1 , p1 ) = (B, B − +1), B). Hier gilt p0 /w0 = 1 und p1 /w1 = (B − 1)/B < 1. Der Algorithmus liefert hier +S = {0} mit Gewinn 1; optimal ist aber Sopt = {1} mit Gewinn B − 1. + In diesem Fall gilt OP T (I)/GA(I) = B − 1. D.h. im allgemeinen Fall ist die Rate +unbeschränkt. + +Hinweis: Man beachte, dass GA in dem obigen Beispiel das Element mit maximalem +Profit pmax ignoriert hat. Aus diesem Grunde betrachten wir eine leichte Modifikation +von GA, die sofort hilft die Approximationsgüte zu beschränken. + +Algorithmus MGA(I = ((w0 , p0 ), . . . , (wn−1 , pn−1 ), B)) + 1 berechne Lösung S1 mit Algorithmus GA, + 2 berechne Lösung S2 = {j} mit pj = maxi∈{0,...,n−1} pi , + 3 wähle Lösung S ∈ {S1 , S2 } mit größerem Gewinn. + 4 return(S). + + + + 233 + 7 Approximative Algorithmen + + + Es sei M GA(I) der Gesamtgewinn der durch Algorithmus M GA berechneten Lösung +S ⊆ {0, . . . , n − 1} und OP T (I) der Gesamtgewinn einer optimalen Lösung zur Eingabe +I. + +Satz 7.18. Die Worst-Case Güte von Algorithmus M GA ist 2 (d.h. OP T (I)/M GA(I) ≤ +2 für alle Eingaben I). + +Beweis. Betrachte relaxierte Version des Rucksackproblems, bei dem Gegenstände frak- +tional genommen werden dürfen. Es sei OP Tf (I) der maximale Wert einer fraktionalen +Lösung x∗ . Es gilt dann OP T (I) ≤ OP Tf (I), da der Lösungsraum sich vergrößert und +ein Max-Problem vorliegt. + Eine fraktionale Lösung x∗ von max nj=1 xj pj mit j xj wj ≤ B und xj ∈ [0, 1] für + P P +alle j = 1, . . . , n nimmt Gegenstände in der sortierten Reihenfolge + + p1 /w1 ≥ p2 /w2 ≥ . . . ≥ pn /wn , + +wobei die ersten k Gegenstände ganzzahlig (d.h. x∗j = 1) und der letzte Gegenstand +fraktional mit x∗k+1 ∈ [0, 1) gewählt wird. +Beachte: Es gilt kj=1 wj ≤ B und k+1 j=1 wj > B. Insgesamt haben wir + P P Pk+1 ∗ + j=1 xj wj = B. +Falls wir einen fraktionalen Gegenstand haben mit x∗k+1 > 0 ist, so gilt kj=1 wj < B. + P + + Im Folgenden zeigen wir, dass eine optimale fraktionalen Lösung die obige Form hat. +Angenommen, wir haben ein Teilstück eines Gegenstands ` > k + 1 der Länge x∗` w` im +Rucksack mit p` /w` < pk+1 /wk+1 und x∗` > 0. Dann muss ein anderes Stück (1 − x∗i )wi +mit i ≤ k +1 und x∗i < 1 im Rucksack fehlen (wegen den obigen Bedingungen). Betrachte +die minimale Länge length = min{(1 − x∗i )wi , x∗` w` }. Das folgende Bild veranschaulicht +die genaue Situation: + pj + wj (1 − x∗i )wi x∗` w` Rucksack: grün + + y` w` + + ... ... ... + ... + + 1 2 i k ` n wj + + austauschen + Dann gilt length = yi wi = y` w` mit yi ∈ (0, 1 − x∗i ] und y` ∈ (0, x∗` ] und pi /wi ≥ +pk+1 /wk+1 > p` /w` . +Frage: Wie bestimmt man yi , y` ? + Man beachte, dass length = min{(1 − x∗i )wi , x∗` w` } > 0. + +Fall 1: (1 − x∗i )wi = length. Hier setze yi = (1 − x∗i ) und y` = length/w` . + +Fall 2: x∗` w` = length. Hier setze y` = x∗` und yi = length/wi . + + + +234 + 7.2 Rucksackproblem + + + Ersetze nun die Länge y` w` in der Lösung durch yi wi . Der Profitgewinn durch diesen +Austausch ist dann yi pi − y` p` und es gilt: + + yi pi − y` p` > yi (p` /w` )wi − y` p` + = yi wi p` /w` − y` p` + = y` w` p` /w` − y` p` + = y` p` − y` p` = 0. + +D.h. der Profitgewinn ist echt positiv und damit kann ein solcher obiger Fall nicht +auftreten. + Nun kommen wir zur Abschätzung der Approximationsgüte: Da k + 1 das erste Item +ist, dass der Algorithmus GA nicht nimmt, gilt + + OP Tf (I) ≤ GA(I) + pk+1 . + +Der Algorithmus M GA berechnet Lösung mit Wert GA(I) oder pmax ≥ pk+1 (bzw. sogar +genauer mit Wert max{GA(I), pmax }). Daher gilt + + OP T (I) ≤ OP Tf (I) ≤ GA(I) + pk+1 + ≤ GA(I) + pmax ≤ 2 max{GA(I), pmax } + = 2 · M GA(I) + + + +Bemerkung 7.19. (a) Der Algorithmus M GA läuft in Zeit O(n log n). + + (b) Durch eine Modifikation von M GA können wir auch einen approximativen Algo- + rithmus angeben mit Worst Case Güte 2, der in O(n) Zeit läuft. + +Beweis. Wir verwenden als Idee zur Laufzeitverbesserung (b) hierzu den Median-Algorithmus +um das sogenannte Split-Item k + 1 in O(n) Zeit zu finden. + + Der folgende Algorithmus verallgemeinert die Idee vom MGA Algorithmus, in dem +er alle k-elementigen Teilmengen durchprobiert, in den Rucksack vorplaziert, wenn das +möglich ist, und mit Hilfe vom GA Algorithmus die Restkapazität auffüllt. +Algorithmus von Sahni + +Gegeben: Parameter k ∈ {0, . . . , n}. +Idee: Für jedes k definiere Algorithmus Ak wie folgt: + + (1) wähle eine Teilmenge S mit |S| ≤ k Gegenständen, die am Anfang im Rucksack + liegen, + + (2) wende Algorithmus GA auf die übrigen Elemente an und füge die ausgewählten + Gegenstände zusätzlich in den Rucksack ein. + + + + 235 + 7 Approximative Algorithmen + + +Wende (1) + (2) auf alle möglichen Teilmengen S mit |S| ≤ k an und wähle Lösung mit +maximalem Gewinn. + Die Technik nennt man k-Enumeration. Es sei Ak (I) der Gewinn der Lösung, den +der Algorithmus Ak von Sahni bezüglich einer Eingabe I und Parameter k ∈ {0, . . . , n} +produziert. + +Satz 7.20. Für alle k ≥ 1 gilt: Ak hat Güte ≤ 1 + 1/k und Laufzeit TAk (n) = O(nk+1 ). + +Beweis. siehe Vorlesung Effiziente Algorithmus. + + Idee für ein schnelleres Verfahren: Konstruiere aus dem exakten Algorithmus +Knapsack aus Kapitel 2 zum Rucksackproblem einen polynomiellen approximativen +Algorithmus. + Erinnerung: Das dynamische Programm zum Rucksackproblem hat eine nicht-polynomielle +(bzw. eine sogenannte pseudo-polynomielle) Laufzeit O(n2 ·pmax ), wobei pmax = max{pi |0 ≤ +i ≤ n − 1}. Hierbei könnte pmax exponentiell in der Eingabegröße sein; siehe auch Be- +merkung 2.20. + Idee zu Algorithmus AK + + (1) konstruiere eine Eingabe IK mit Gewinnen p0i = bpi /Kc (alle anderen Größen wi + und B bleiben gleich). + + (2) wende Algorithmus Knapsack auf Eingabe IK an. + + Der Algorithmus AK berechnet eine Lösung S ⊆ {0, . . . , n−1} mit Größe + P + i∈S wi ≤ B +und maximalem Gewinn i∈S p0i . + P + Wir entwickeln aus AK ein Verfahren A durch die Wahl + pmax + K= . + (1/ + 1)n + +Satz 7.21. Der Algorithmus A läuft in Zeit O(n3 /) und hat eine Worst-Case Güte +OP T (I)/A (I) ≤ 1 + . + + Man nennt eine solche Familie von Algorithmen (A ) ein vollständiges polynomielles +Approximationsschema (FPTAS). + +Beweis. (a) zur Laufzeit: Für die skalierten Profite gilt: p0max = maxi p0i = maxi bpi /Kc = +maxi bpi /pmax (1/ + 1)nc = b(1/ + 1)nc = O(n/). + Der Algorithmus Knapsack angewandt auf die skalierte Instanz hat dann eine Laufzeit +von + O(n2 p0max ) = O(n2 n/) = O(n3 /). + (b) zur Güte: Es gilt zunächst + + OP T (I) ≤ K · OP T (IK ) + K · n (1) + + Da p0i = bpi /Kc ≥ pi /K − 1, haben wir pi ≤ K · (p0i + 1) und deswegen haben wir + + + +236 + 7.2 Rucksackproblem + + + + pi ≤ i∈Iopt (Kp0i + K) + P P + OP T (I) = i∈Iopt + 0 + P + ≤ i∈Iopt Kpi + K · n ≤ K · OP T (IK ) + K · n + + Beachte, dass Iopt eine zulässige Lösung von IK ist und (IK )opt i.a. ein höheren Gewinn +als Iopt bezüglich Instanz IK hat. Daneben gilt: + + K · OP T (IK ) ≤ AK (I) (2) + Da p0i ≤ pi /K gilt + + K · OP T (IK ) = K · i∈(IK )opt p0i ≤ K · i∈(IK )opt pi /K + P P + P + = i∈(IK )opt pi ≤ AK (I). + + Aus (1) + (2) folgt + OP T (I) ≤ AK (I) + K · n. +Da alle wi ≤ B (ansonsten kann Gegenstand i aus der Instanz gelöscht werden), gilt +OP T (I) ≥ pmax = maxi pi . + Wir erhalten + OP T (I) AK (I) + Kn + ≤ + A (I) AK (I) + Kn + =1+ + AK (I) + Kn + ≤1+ + OP T (I) − Kn + Kn + ≤1+ + pmax − Kn + = 1 + . + +Die letzte Gleichung gilt, da + pmax + Kn (1/+1)n · n + = pmax + pmax − Kn pmax − (1/+1)n n + 1 1 + (1/+1) 1/+1 + = 1 = 1/+1−1 + 1 − (1/+1) + 1/+1 + 1 + = = . + 1/ + +Weitere Approximationsschema + + • O(n2 /) (Ibarra und Kim 1975). + + • O(n log(1/) + 1/4 ) (Lawler 1979). + + + + 237 + 7 Approximative Algorithmen + + + • O(n min{log n, log(1/)} + 1/2 log(1/) min{n, 1/ log(1/)}) (Kellerer, Pferschy + 2004). + + • Õ(n + (1/)5/2 ) (Rhee 2015). + + • Õ(n + (1/)12/5 ) (Chan 2018). + + • Õ(n + (1/)9/4 ) (Jin 2019). + +In der Õ Notation vernachlässigt man log(.)-Terme in der Laufzeit. + +Bemerkung 7.22. Es gibt kein FPTAS mit einer Laufzeit O((n + 1/)2−δ ), außer +(min, +) Convolution hat einen Algorithmus mit einer subquadratischen Laufzeit (Cygan +u.a. 2017 bzw. Künnemann u.a. 2017). + + +7.3 Scheduling +Der Begriff Scheduling bescheibt die Problematik der Erstellung eines Ablaufplans bzw. +Schedules, der Prozessen zeitlich begrenzt Ressourcen, wie etwa Maschinen bzw. Prozes- +soren oder Speicher, zuteilt. Dabei untersucht man die Optimierung bzw. Approximation +diverser Zielfunktionen. Eine klassische Variante wird formal als P ||Cmax angegeben und +lässt sich wie folgt formulieren. +Gegeben: + n Jobs J = {J1 , . . . , Jn } mit Ausführungszeiten p1 , . . . , pn ∈ N und m identische +Maschinen. +Gesucht: + Partition von J in m Teilmengen B1 , . . . , Bm mit minimaler maximaler Last + X + Cmax = max pj . + 1≤i≤m + Jj ∈Bi + + +Den Wert Cmax nennt man auch den Makespan des Schedules. Das Problem wird be- +zeichnet durch P ||Cmax bzw. P m||Cmax , wenn die Anzahl der Maschinen m konstant ist. +Im Folgenden betrachten wir eine Instanz für das Schedulingproblem P ||Cmax : + + Tab. 7.1: Schedulinginstanz + + Jj J1 J2 J3 J4 J5 + m=2 + pj 1 2 1 3 1 + + Für diese Instanz mit zwei Maschinen können wir etwa die beiden folgenden Ablauf- +pläne/Schedules als zulässig erkennen: + + • B1 = {J1 , J2 , J5 }, B2 = {J4 , J5 } mit Makespan 4. + + + +238 + 7.3 Scheduling + + + M1 J1 J2 J3 + M2 J4 J5 + Zeit + 0 1 2 3 4 5 + + + • B1 = {J1 , J3 , J4 }, B2 = {J2 , J5 } mit Makespan 5. + + M1 J1 J3 J4 + M2 J2 J5 + Zeit + 0 1 2 3 4 5 + + + Man erkennt leicht, dass der erste Ablaufplan ein optimaler Ablaufplan ist. Der op- +timale Makespan ist also 4. Wie schwer ist nun dieses Problem? Der folgende Satz gibt +uns Auskunft. + +Satz 7.23. Das Schedulingproblem auf identischen Maschinen P m||Cmax ist NP-vollständig +sogar für m = 2 Maschinen. + +Beweis. Dies folgt durch eine Reduktion des Partitionsproblems. + + Man mache sich klar, dass daraus unmittelbar auch die NP-Vollständigkeit von P ||Cmax +folgt. Wir wollen nun einen ersten Approximationsalgorithmus studieren, den wir List- +Scheduling nennen. + +Algorithmus ListScheduling(L=(J1 , . . . , Jn ),m) + 1 for i=1 to m do + 2 Ei = 0; Bi = ∅; + 3 od + 4 for j=1 to n do + 5 wähle Job Jj aus Liste L; + 6 wähle Maschine Mi mit minimaler Last Ei ; + 7 Bi = Bi ∪ {Jj }; + 8 Ei = Ei + pj ; + 9 od + + Es sei LS(I) die Länge eines List Schedules und OP T (I) die Länge eines optimalen +Schedules zur Eingabe I. + +Satz 7.24. (a) Für alle Eingaben I = (L, m) gilt LS(I)/OP T (I) ≤ 2 − 1/m. + + (b) Es existiert eine Eingabe I ∗ mit LS(I ∗ ) = (2 − 1/m)OP T (I ∗ ). Der Algorithmus + hat also eine multiplikative Güte bzw. absolute Worst Case Güte von 2 − 1/m. + + + + 239 + 7 Approximative Algorithmen + + +Beweis. Zu (a): O.B.d.A. hat Maschine M1 nach der Zuordnung die höchste Last L = + j∈B1 pj (ansonsten nummeriere die Maschinen um). Es sei Jk der letzte Job, der auf +P +Maschine M1 fertig wird. Dann haben alle Maschinen eine Last Li ≥ L − pk . Zu dem +Zeitpunkt, wo Jk der Maschine M1 zugeordnet wurde, hatte M1 die kleinste Last L − pk . +Wir illustieren dies in Abb. 7.1. + + M1 Jk + M2 + M3 + .. + . + Mm + Zeit + 0 L − pk L + + Abb. 7.1: Maschine M1 hat mit Jk bzw. ohne Jk die größte Last bzw. kleinste Last + + Daraus folgt sofort die folgende Abschätzung: + n + X + pi ≥ m(L − pk ) + pk . + i=1 + Daneben gilt (da die gesamte Last Ppi auf m Maschinen verteilt werden muss und + P +mindestens eine Maschine eine Last ≥ pi /m hat): + n + X + pi /m ≤ OP T (I). + i=1 + +Da nun LS(I) = L ist, gilt + n + X + OP T (I) ≥ pi /m ≥ m(L − pk )/m + pk /m + i=1 + = L − (1 − 1/m)pk = LS(I) − (1 − 1/m)pk + +Da OP T (I) ≥ pk folgt nun OP T (I) ≥ LS(I) − (1 − 1/m)OP T (I) bzw. + + LS(I) ≤ (2 − 1/m)OP T (I). + +Zu (b): zur Übung. + + Wir haben gesehen, dass ein erstaunlich einfacher Algorithmus schon eine Güte von 2 +garantieren kann. Allerdings fällt auf, dass die beliebige Wahl des nächsten Jobs (in Zeile +5) mitunter schlecht sein kann. Es scheint besser zu sein, die Jobs absteigend ihrer Aus- +führungszeit nach zu plazieren, was uns zum folgenden Algorithmus LPT Scheduling +führt. + + + +240 + 7.3 Scheduling + + +Algorithmus LPT Scheduling(I=(J,,m)) + 1 sortiere die Jobs in J so, dass p1 ≥ p2 . . . ≥ pn ; + 2 wende den List Scheduling Algorithmus an auf I=(L,m) mit L=(J1 , . . . , Jn ) + + Es sei LP T (I) die Länge eines LPT Schedules bezüglich einer Eingabe I = (J, m). + +Satz 7.25. Der LPT Algorithmus hat eine absolute Worst Case Rate von 4/3 − 1/(3m); +d.h. + LP T (I) ≤ (4/3 − 1/(3m))OP T (I). (∗) + +Beweis. Annahme: Es existiert eine Jobmenge J und Maschinenzahl m, die die Be- +hauptung (∗) nicht erfüllt. Der Satz gilt sofort für m = 1 (in diesem Fall gilt nämlich +LP T (I) = OP T (I)). Daher nehmen wir an: m ≥ 2 und n ist minimal (d.h. wir haben +ein minimales Gegenbeispiel). +Annahme: Es existiert ein Job Jr mit r < n, für den die Fertigstellungszeit fr = +LP T (I = (J, m)) gilt. Betrachte J 0 = {J1 , . . . , Jr } mit Liste L0 = (J1 , . . . , Jr ). Dann gilt +LP T (I = (J, m)) = LP T (I 0 = (J 0 , m)) und der optimale Wert OP T (I 0 ) ≤ OP T (I). +Damit haben wir + LP T (I 0 ) LP T (I) + ≥ > 4/3 − 1/(3m) + OP T (I 0 ) OP T (I) +ein kleineres Gegenbeispiel gefunden. Widerspruch zur Minimalität von n. + Es gilt also fk < LP T (I = (J, m)) für alle k < n. Desweiteren haben wir die Unglei- +chungen: + n + 1 X + OP T (I) ≥ pi (1) + m + i=1 + n−1 + (2) + X + pi ≥ msn = m(LP T (I) − pn ), + i=1 + +wobei sn der Startzeitpunkt von Job Jn ist. + + pn + Jn + + m Jobs J1 , . . . , Jn−1 + + + Zeit + 0 sn LPT(J) + + + Beachte: Alle Maschinen sind bis zum Zeitpunkt sn voll beschäftigt (d.h. keine Ma- + + + + 241 + 7 Approximative Algorithmen + + +schine ist idle vor dem Zeitpunkt sn ). + + LP T (I) sn + pn + = + OP T (I) OP T (I) + n−1 + pn 1 X + ≤(2) + pi + OP T (I) mOP T (I) + i=1 + n + (m − 1)pn 1 X + = + pi + mOP T (I) mOP T (I) + i=1 + (m − 1)pn + ≤(1) + 1. + mOP T (I) + +Da (∗) für I nicht gilt, folgt + + (m − 1)pn LP T (I) + 1+ ≥ > 4/3 − 1/(3m). + mOP T (I) OP T (I) + +Daraus folgt + (m − 1)pn m−1 + > 1/3 − 1/(3m) = + mOP T (I) 3m +bzw. + OP T (I) + pn > . + 3 +(d.h. der kleinste Job Jn ist größer als OP T (I)/3). Daraus folgt, dass keine Maschine in +einem optimalen Schedule mehr als 2 Jobs ausführen kann. + Als nächstes transformieren wir einen optimalen Schedule Dopt für solch eine Jobmenge +mit pn > OP T (I)/3 in einen Schedule D̄opt , der zusätzliche Eigenschaften erfüllt. +Transformation I.1 für Dopt . Gegeben sei ein Schedule Dopt mit Makespan Cmax und +zwei Jobs Ji und Ji0 auf einer Maschine und zwei Jobs Jj und Jj 0 auf einer anderen +Maschine, wobei pi > pj und pi0 > pj 0 ist. + + + + + Ji Ji0 + + + Jj Jj 0 + + + + Zeit + 0 + +Wenn wir Ji0 und Jj 0 vertauschen, so erhalten wir den folgenden Schedule Dopt + 0 mit +Makespan Cmax ≤ Cmax : + 0 + + + + +242 + 7.3 Scheduling + + + + + Ji Jj 0 + + + Jj Ji0 + + + + Zeit + 0 + +Transformation I.2 für Dopt . Gegeben sei ein Schedule Dopt mit Makespan Cmax und +zwei Jobs Ji und Ji0 auf einer Maschine und ein Job Jj auf einer anderen Maschine, +wobei pi > pj ist. + Wenn wir den Job Ji0 auf die andere Maschine verschieben, so erhalten wir einen +Schedule Dopt + 0 ebenso mit Makespan Cmax + 0 ≤ Cmax : + + + + + Ji Ji0 + + verschieben + Jj + + + + Zeit + 0 + +Transformation II für Dopt . Gegeben sei ein Schedule Dopt mit Makespan Cmax und +zwei Jobs Ji und Ji0 auf einer Maschine mit pi < pj . In diesem Fall vertauschen wir die +Reihenfolge beider Jobs auf der entsprechenden Maschine: + + + + + austauschen + Ji Jj + + + + Zeit + 0 + +Beachte: Keine der Transformationstypen I.1, I.2 oder II erhöht den Makespan. + + + + 243 + 7 Approximative Algorithmen + + +Definition 7.26 (Lastfunktion). Bei gegebenen Lastwerten Li auf den Maschinen Mi +in einem Schedule D ist die Lastfunktion L(D) = 1≤i pj , so +folgt + + pj 0 ≥ pi0 , pi ≤ pk , pi ≥ pi0 , (∗∗) + pj ≤ pk , pj ≥ pj 0 . + + Durch Umordnen der Maschinen in D∗ erhalten wir einen Schedule D̄opt : + + + +244 + 7.3 Scheduling + + + Jk1 + Jk2 + + + + + Jks + Ji1 Ji01 + Ji2 Ji02 + + + + + Jit Ji0t + Zeit + 0 + + Hierbei gelten die Ungleichungen pk1 ≥ . . . ≥ pks sowie pi1 ≥ . . . ≥ pit und wegen (∗∗) +haben wir p0i1 ≤ . . . ≤ p0it und pks ≥ pi1 sowie pit ≥ p0it . Damit haben wir die folgende +Ordnung der Jobs: + pk1 ≥ . . . ≥ pks ≥ pi1 ≥ . . . ≥ pit ≥ p0it ≥ . . . p0i1 . + Der Schedule D̄opt ist nun äquivalent zu einem List Schedule DL bei gegebener Lis- +te L = (J1 , . . . , Jn ) mit p1 ≥ . . . ≥ pn (bis auf Vertauschen von Jobs mit gleicher +Ausführungszeit). Daraus folgt aber + + LP T (I = (J, m)) = OP T (I = (J, m)). + Wegen LP T (I)/OP T (I) > 4/3 − 1/(3m) erhalten wir nun aber einen Widerspruch +(d.h. es kann kein Gegenbeispiel für die Ungleichung (∗) existieren). + +Bemerkung 7.28. Für jede Anzahl m von Maschinen existiert eine Eingabe Im + ∗ = + +(J ∗ , m) mit + ∗) + LP T (Im 4 1 + ∗) + = − . + OP T (Im 3 3m +Beweis. zur Übung. + +Satz 7.29. Für jede Genauigkeit  > 0 gibt es einen approximativen Algorithmus A für +P ||Cmax mit A (I) ≤ (1 + )OP T (I) und Laufzeit 2O(1/ log (1/)) + O(n). + 2 + + + +Bemerkung 7.30. Eine Familie (A ) von solchen Algorithmen nennt man ein (effizi- +entes) polynomielles Approximationsschema (EPTAS). +Details: siehe Vorlesung Effiziente Algorithmen. + + + + 245 + 7 Approximative Algorithmen + + +Übungsaufgaben +Übung 7.31. Zeigen Sie: + + • Ein zusammenhängender ungerichteter Graph G = (V, E) enthält eine Eulertour + genau dann, wenn jeder Knoten in V geraden Knotengrad hat. + + • Ein stark zusammenhängender gerichteter Graph G = (V, E) enthält eine Eulertour + genau dann, wenn für jeden Knoten v ∈ V gilt, dass din (v) = dout (v). + + + + +246 + \ No newline at end of file diff --git a/uni/aak/alle_hausaufgaben.pdf b/uni/aak/alle_hausaufgaben.pdf new file mode 100644 index 0000000..2bd9ce0 --- /dev/null +++ b/uni/aak/alle_hausaufgaben.pdf @@ -0,0 +1,7959 @@ +%PDF-1.3 +% +1 0 obj +<< /Metadata 3 0 R /Pages 4 0 R /Type /Catalog >> +endobj +2 0 obj +<< /Author () /CreationDate (D:20250713031408+02'00') /Creator (TeX) /ModDate (D:20250713031408+02'00') /Producer (pdfTeX-1.40.25) >> +endobj +3 0 obj +<< /Subtype /XML /Type /Metadata /Length 793 >> +stream + + + + + + + + +endstream +endobj +4 0 obj +<< /Count 21 /Kids [ 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 10 0 R 11 0 R 12 0 R 13 0 R 14 0 R 15 0 R 16 0 R 17 0 R 18 0 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/Info 2 0 R /Root 1 0 R /Size 651 /ID [<58d2f731fa4fb722ecc943c307d52835><58d2f731fa4fb722ecc943c307d52835>] >> +startxref +1312234 +%%EOF diff --git a/uni/aak/alle_hausaufgaben.txt b/uni/aak/alle_hausaufgaben.txt new file mode 100644 index 0000000..6e3f255 --- /dev/null +++ b/uni/aak/alle_hausaufgaben.txt @@ -0,0 +1,817 @@ + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 09.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 9 + + +Definition 1 (Vertex Cover (VC)). Für einen (ungerichteten) Graphen G = (V, E) ist ein Vertex +Cover eine Menge von Knoten C ⊆ V , sodass u ∈ C oder v ∈ C für alle { u, v } ∈ E gilt, also ist +für alle Kanten mindestens ein Endpunkt in C. +Für das V ERTEX C OVER Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gegeben und +es wird gefragt, ob ein Vertex Cover mit Größe (Anzahl der Knoten) höchstens k in G existiert. + +Definition 2 (Feedback Vertex Set (FVS)). Für einen gerichteten Graphen G = (V, E) ist ein +Feedback Vertex Set eine Menge von Knoten X ⊆ V , sodass G \ X kreisfrei ist. +Für das F EEDBACK V ERTEX S ET Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gege- +ben und es wird gefragt, ob ein Feedback Vertex Set mit Größe (Anzahl der Knoten) höchstens +k in G existiert. + + + 1 6 2 + + + 5 + 8 7 + 3 + 4 + Abbildung 1: Beispiel Graph. + +Hausaufgabe 9.1 (Nichtdeterministischer Algorithmus für VC) (5 Punkte) + 1. Finden Sie ein minimales Vertex Cover in Abbildung 1. Für ein Vertex Cover ist die + Orientierung der Kanten nicht relevant. + 2. Zeigen Sie V ERTEX C OVER ∈ NP, indem Sie einen nichtdeterministischen Algorithmus + angeben, der V ERTEX C OVER in Polynomialzeit löst. Begründen Sie die Korrektheit und + die Laufzeit von Ihrem Algorithmus. + +Lösung. 1. Ein minimales Vertex Cover ist zum Beispiel { 3, 5, 6, 7, 8 }. + 2. Unser nichtdeterministischer Algorithmus geht wie folgt vor: Für jeden Knoten wird + nichtdeterministisch entschieden, ob dieser im Vertex Cover sein soll. Nachdem dies für + alle Knoten entschieden wurde, wird überprüft, ob nicht mehr als k Knoten gewählt wur- + den und ob für jede Kante mindestens ein Endpunkt im Vertex Cover liegt. + Existiert ein Vertex Cover im Graphen, existiert auch eine Reihe von nichtdeterministi- + schen Entscheidungen, die diese Teilmenge von Knoten auswählt. Diese Teilmenge wird + dann auch von dem Algorithmus akzeptiert. Damit ist der Algorithmus korrekt. + Beide diese Schritte können naiv in Polynomialzeit erledigt werden, also löst dieser Al- + gorithmus das Vertex Cover Problem in Polynomialzeit. + + +Bewertung 1 Punkt für das Beispiel und 4 Punkte für den Algorithmus, dabei 2 Punkte für +die Beschreibung und jeweils 1 Punkt für Korrektheit und Laufzeit. + +Hausaufgabe 9.2 (Polynomieller Verifizierer für FVS) (5 Punkte) + 1. Finden Sie ein minimales Feedback Vertex Set in Abbildung 1. + 2. Zeigen Sie F EEDBACK V ERTEX S ET ∈ NP, indem Sie einen polynomiellen Verifizierer + für F EEDBACK V ERTEX S ET angeben. Begründen Sie die Korrektheit und die Laufzeit + von Ihrem Verifizierer. + +Lösung. 1. Ein minimales FVS ist zum Beispiel { 5, 6 }. + 2. Das Zertifikat hat die Form einer Teilmenge X ⊆ V , die beschreibt, welche Knoten das + Feedback Vertex Set bilden sollen. Zum Beispiel über ein Bit pro Knoten. Nun wird + überprüft, ob |X| ≤ k gilt und ob G \ X azyklisch ist. + Hat der Graph ein Feedback Vertex Set von Größe maximal k, gibt es ein Zertifikat, wel- + ches dieses beschreibt. Dieses Zertifikat wird dann von unserem Verifizierer akzeptiert, + damit ist der Verifizierer korrekt. + Die Größe von X zu überprüfen ist naiv in Polynomialzeit möglich. Für den zweiten Test + bietet sich der Algorithmus zur topologischen Sortierung aus der Vorlesung an, welcher in + Polynomialzeit eine topologische Sortierung findet, wenn der gegebene Graph azyklisch + ist. Insgesamt ist dieser Verifizierer korrekt und auch ein polynomieller Verifizierer. + + +Bewertung 1 Punkt für das Beispiel und 4 Punkte für den Verifizierer, dabei 2 Punkte für die +Beschreibung und jeweils 1 Punkt für Korrektheit und Laufzeit. + + + + +Abgabe: 16.06.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 9 + Marek Lenczewski + Matrikelnummer: 1025252 + + 22. Juni 2025 + + +Hausaufgabe 9.1 +Teil 1 +Ein minimales Vertex Cover ist C = {2, 7, 5, 4, 1}. +Die Minimale Knotenanzahl ist 5. + +Teil 2 +Der Algorithmus arbeitet wie folgt: + + • Zuerst wird für jeden Knoten v ∈ V nichtdeterministisch entschieden, ob dieser in das + Vertex Cover aufgenommen wird. Dabei wird drauf geachtet, dass die Anzahl k nicht + übersteigt. + + • Danach wird geprüft, ob für jede Kante {u, v} ∈ E entweder v ∈ C oder u ∈ C oder beides + zutrifft. + + • Falls dies für alle Kanten gilt, akzeptieren wir. Andernfalls lehnen wir ab. + + Existiert ein Vertex Cover der Größe höchstens k, dann gibt es auch eine, nichtdeterminis- +tisch ausgewählte, Knotenmenge, die dazu passt. Von dieser werden alle Kanten abgedeckt und +der Algorithmus akzeptiert. Falls es kein Vertex Cover gibt, dann lehnt der Algorithmus ab. + + Die Auswahl der Knoten und die Prüfung der Kanten läuft in polynomieller Zeit, also ist dies +ein polynomieller nichtdeterministischer Algorithmus, der VERTEX COVER löst und damit ist +V ERT EXCOV ER ∈ N P . + + +Hausaufgabe 9.2 +Teil 1 +Ein minimales Feedback Vertex Set ist X = {1, 5} + +Teil 2 +Der Verifizierer arbeitet wie folgt: + + • Es wird eine Teilmenge X ⊆ V übergeben, die das Feedback Vertex Set darstellen soll. + + • Dann wird geprüft, ob |X| ≤ k gilt und es wird abgelehnt, falls es nicht gilt. + + • Danach wird ein Graph G′ = G \ X konstruiert, in dem alle Knoten aus X und alle an X + angrenzenden Kanten entfernt wurden. + + 1 + • Auf dem Graphen G′ wird eine Tiefensuche durchgeführt, dabei werden Knoten während + der Bearbeitung grau markiert und beim Backtracking schwarz. + + • Wird eine Kante zu einem grauen Knoten entdeckt, dann wird abgelehnt, da dies einen + Kreis anzeigt. + + • Falls die Tiefensuche den gesamten Graphen durchläuft ohne abzulehnen, dann wird ak- + zeptiert. + + Existiert ein Feedback Vertex Set der Größe höchstens k, dann gibt es auch ein Zertifikat, das +genau diese Knotenmenge beschreibt. Alle Teilmengen, die die Tiefensuche überstehen, werden +von dem Verifizierer akzeptiert und sonst abgelehnt. + + Die Prüfung der übergebenen Teilmenge und die Konstruktion des Graphen laufen in polyno- +mieller Zeit. Ebenfalls die Tiefensuche läuft in polynomieller Zeit, somit ist auch der Verifizierer +polynomiell und verifiziert die Lösung für das FEEDBACK VERTEX SET in polynomieller +Zeit. Also gilt F EEDBACKV ERT EXSET ∈ N P . + + + + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 16.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 10 + + +Definition 1 (Feedback Vertex Set (FVS)). Für einen gerichteten Graphen G = (V, E) ist ein +Feedback Vertex Set eine Menge von Knoten X ⊆ V , sodass G \ X kreisfrei ist. +Für das F EEDBACK V ERTEX S ET Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gege- +ben und es wird gefragt, ob ein Feedback Vertex Set mit Größe (Anzahl der Knoten) höchstens +k in G existiert. + +Hausaufgabe 10.1 (F EEDBACK V ERTEX S ET ist NP-vollständig) (5 Punkte) +Zeigen Sie, dass F EEDBACK V ERTEX S ET NP-vollständig ist. +Hinweis: Reduzieren Sie V ERTEX C OVER auf F EEDBACK V ERTEX S ET. Wenn ein Vertex Cover +aus einem Graphen entfernt wird, hat dieser keine Kanten mehr. + +Lösung. F EEDBACK V ERTEX S ET ∈ NP ist bekannt aus der letzten Hausaufgabe. Nun zeigen +wir eine Reduktion von V ERTEX C OVER auf F EEDBACK V ERTEX S ET. Betrachte die folgende +Reduktion: +Für eine gegebene V ERTEX C OVER Instanz (G = (V, E), k) erzeuge den Graphen G′ = (V, E ′ ) +mit E ′ := { (u, v), (v, u) | {u, v} ∈ E }. Gebe (G′ , k) als F EEDBACK V ERTEX S ET Instanz aus. +Korrektheit: +⇒ (Falls (G, k) eine Ja-Instanz von V ERTEX C OVER ist, dann ist (G′ , k) eine Ja-Instanz von +F EEDBACK V ERTEX S ET.): +Sei C ⊆ V ein Vertex Cover mit Größe kleiner oder gleich k. Dann enthält G \C keine Kanten. +Damit hat G′ \C ebenfalls keine Kanten und somit keine Kreise. +⇐ (Falls (G′ , k) eine Ja-Instanz von F EEDBACK V ERTEX S ET ist, dann ist (G, k) eine Ja-Instanz +von V ERTEX C OVER.): +Sei X ⊆ V ′ ein Feedback Vertex Set mit Größe kleiner oder gleich k. Dann ist X ein Vertex +Cover in G, da ansonsten eine Kante {u, v} ∈ E existiert, für die u ∈ + / X ∧v ∈ + / X gilt. Damit wäre +aber (u, v, u) ein Kreis in G′ und X somit kein Feedback Vertex Set. +Laufzeit: Das Erzeugen der Menge E ′ geht in Zeit O(|E|). Insgesamt ist die Reduktion dem- +nach in Laufzeit O(|E|), also in polynomieller Laufzeit, durchführbar. +Also existiert eine polynomielle Reduktion von V ERTEX C OVER auf F EEDBACK V ERTEX S ET. +Da V ERTEX C OVER nach den aktuellen Präsenzaufgaben NP-vollständig ist und +F EEDBACK V ERTEX S ET ∈ NP gilt, ist somit F EEDBACK V ERTEX S ET NP-vollständig. + Bewertung 2 Punkte für die Konstruktion, 1 Punkt für den 1. Fall, 1 Punkt für den 2. Fall, +0.5 Punkte für die Laufzeit, 0.5 Punkte für die Beweisführung im Ganzen + +Hausaufgabe 10.2 (Turingmaschinen) (5 Punkte) +Entwerfen Sie eine Turingmaschine für die Sprache L = { w ∈ Σ∗ | w ist ein Palindrom } über +einem gegebenen Alphabet Σ und geben Sie die Laufzeit Ihrer Turingmaschine an. Begründen +Sie dabei, warum Ihre Turingmaschine korrekt ist und die angegebene Laufzeit hat. + +Lösung. Die Turingmaschine sieht folgendermaßen aus: + (1) Falls kein Wort auf dem Band steht, akzeptiere. + (2) Falls nur ein Buchstabe auf dem Band steht, akzeptiere. + (3) Lese den ersten Buchstaben und bewege danach den Kopf zum letzten Buchstaben. Falls + die beiden Buchstaben verschieden sind, verwerfe. Ansonsten ersetze beide Buchstaben + durch ein neues Symbol x. Ersetze dafür zuerst den letzten Buchstaben und bewege dann + den Kopf zum ersten Buchstaben. + (4) Gehe zu Schritt (1). +Korrektheit: Die Turingmaschine vergleicht immer den ersten und den letzten Buchstaben. +Wenn die Buchstaben nicht gleich sind, ist das Wort kein Palindrom und es wird verworfen. +Ansonsten werden die Buchstaben rausgestrichen (siehe Schritt (3)). Diese Prozedur wird so- +lange wiederholt, bis man in der Mitte des Wortes angekommen ist und entweder nur noch ein +Buchstabe oder kein Buchstabe vorhanden ist. In beiden Fällen ist das Wort ein Palindrom und +es wird akzeptiert (siehe Schritte (1) und (2)). +Laufzeit: Die Schritte (1) bis (3) benötigen jeweils eine Laufzeit von O(n), wobei n hier die +Länge der Eingabe bezeichnet. Da in jedem Durchlauf der Schritte (1) bis (3) zwei Buchstaben +raus gestrichen werden oder das Wort akzeptiert oder verworfen wird, gibt es maximal ⌊n/2⌋ +Durchläufe, bis nur noch ein oder kein Buchstabe auf dem Band stehen und somit in den Schrit- +ten (1) oder (2) akzeptiert wird. Damit ergibt sich insgesamt eine Laufzeit von O(n2 ). + + +Bewertung 2 Punkte für die Turingmaschine, 1 Punkt für die Korrektheit und 2 Punkte für +die Laufzeit: 1 Punkt Angabe, 1 Punkt Begründung + + + + +Abgabe: 23.06.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 10 + Marek Lenczewski + Matrikelnummer: 1025252 + + 22. Juni 2025 + + +Hausaufgabe 10.1 +Teil 1: FeedbackVertexSet ∈ NP + Zuerst brauchen wir einen polynomiellen Verifizierer, um zu zeigen dass FeedbackVertexSet +∈ NP. + + Der Verifizierer bekommt einen Graphen G und eine Teilmenge X, dieser arbeitet wie folgt: + + • Prüfe, ob |X| ≤ k, sonst ablehnen + + • Den Graphen G′ = G \ X konstruieren + + • Tiefensuche auf G′ durchführen, dabei werden Knoten vom aktuellen Pfad als grau mar- + kiert und nach vollständiger Bearbeitung als schwarz. Wird eine Kante zu einem grauen + Knoten gefunden, zeigt dies einen Kreis an und wir lehnen ab + + • Akzeptieren, wenn die Tiefensuche nichts findet + + Die Schritte 1-4 laufen in polynomieller Zeit, somit läuft der Verifizierer in polynomieller +Zeit. Damit ist F eedbackV ertexSet ∈ N P . + + Teil 2: FeedbackVertexSet ist NP-schwer + Jetzt brauchen wir eine Reduktion von VertexCover auf FeedbackVertexSet, um zu zeigen, +dass F eedbackV ertexSet N P − Schwer ist. + + Konstruktion: Sei dafür ein VertexCover gegeben mit einem Graphen G und einer Zahl k. +Aus G wird ein G′ , wobei jede ungerichtete Kante {u, v} ∈ E durch zwei gerichteten Kanten +(u, v) und (v, u) zu E ′ ersetzt wird. Das k wird einfach übernommen k = k ′ . + + Beweis VertexCover nach FeedbackVertexSet: + + • Sei C ein VertexCover von G mit |C| ≤ k + + • Da C ein VertexCover ist, wird jede Kante mit C verbunden + + • Wenn C entfernt wird, dann hat G \ C keine Kanten mehr + + • Dann hat auch G′ \ C keine Kanten und ist somit Kreisfrei + + • Damit ist C ein FeedbackVertexSet für G′ mit |C| ≤ k ′ + + + + Beweis FeedbackVertexSet nach VertexCover: + + 1 + • Sei X ein FeedbackVertexSet von G′ mit |X| ≤ k ′ + + • Dann ist G′ \ X kreisfrei + + • Angenommen X wäre kein VertexCover von G, dann gäbe es eine Kante {u, v} ∈ E mit + u, v ∈ + /X + + • Somit wären u, v noch in G′ \ X vorhanden + + • Die Kanten (u, v) und (v, u) würden, durch die Konstruktion, einen Kreis erzeugen + + • Das ist ein Widerspruch, somit ist G′ \ X kreisfrei und X ist ein VertexCover + + Da F eedbackV ertexSet ∈ N P und N P − schwer ist, folgt dass F eedbackV ertexSet NP- +Vollständig ist. □ + + +Hausaufgabe 10.2 +Lösung. Die Turingmaschine arbeitet wie folgt: + + 1. Falls das Band leer ist oder nur aus Markierungen besteht, akzeptiere + + 2. Merke das erste unmarkierte Zeichen und ersetze es durch X + + 3. Laufe zum letzten unmarkierten Zeichen oder verwefe + + 4. Falls dieses Zeichen nicht mit dem gemerkten übereinstimmt, verwerfe + + 5. Ersetze es durch X und gehe zurück zum Anfang + + 6. Gehe zu 1 + + Korrektheit: Ein Wort ist genau dann ein Palindrom, wenn es sich von vorne und hin- +ten gleich liest. Dies prüft die Turingmaschine, indem sie Schrittweise die äußeren Buchstarben +verifiziert, bis nur noch einer oder keiner Vorhanden ist, dann akzeptiert ist und wenn eine Un- +stimmigkeit auftritt, dann lehnt sie ab. + + Laufzeit: Die Turingmaschine hat eine Laufzeit von O(n2 ). In jedem Durchlauf macht sie +O(n) Schritte und das mit O(n) Durchläufen. □ + + + + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 23.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 11 + + +Definition 1 (Dreiecksüberdeckung eines Graphen (∆ C OVER)). Gegeben ist ein endlicher un- +gerichteter Graph G = (V, E) und eine Zahl k ∈ N. Dieser Graph hat eine Dreiecksüberdeckung, +wenn eine Menge von Knoten C∆ ⊆ V mit |C∆ | ≤ k existiert, so dass zu jedem Teilgraphen (Drei- +eck) D = (VD , ED ) ⊆ (V, E) mit |VD | = 3 und ED = { {v1 , v2 } ∈ VD ×VD | v1 ̸= v2 } von G gilt, +v ∈ C∆ für mindestens ein v ∈ VD . + +Hausaufgabe 11.1 (10 Punkte) + + + (a) (1 Punkt) Markieren Sie im folgenden Graphen (Abbildung 1 (a)) ein V ERTEX C OVER + der Größe 5. + (b) (1 Punkt) Markieren Sie im folgenden Graphen (Abbildung 1 (b)) eine Dreiecksüberde- + ckung der Größe 5. + (c) (8 Punkte) Zeigen Sie, dass ∆ C OVER NP-vollständig ist. + + + + + (a) (b) + + Abbildung 1: Beispielgraph + + +Lösung. + (a) (b) + + Abbildung 2: Beispielgraph Lösung + + +(c) Wir zeigen zunächst, ∆ C OVER ∈ NP mittels eines polynomiellen Verifizierers. + Der polynomielle Verifizierer erhält als Zertifikat eine Menge von Knoten C∆ ⊆ V . Zunächst + überprüft er, ob |C∆ | ≤ k. Falls dies nicht der Fall ist, verwerfe die Instanz. Ansonsten überprüfe + für alle Teilgraphen D = (VD , ED ) ⊆ (V, E) mit |VD | = 3 und ED = { {v1 , v2 } ∈ VD ×VD | v1 ̸= v2 } + von G, ob für mindestens ein v ∈ VD auch v ∈ C∆ gilt. Falls dies gilt, akzeptiere. Ansonsten ver- + werfe. + Korrektheit: Der Verifizierer überprüft die geforderten Eigenschaften. Falls ein Graph eine Drei- + ecksüberdeckung besitzt, so existiert ein entsprechendes Zertifikat C∆ und der Verifizierer ak- + zeptiert die Instanz. Falls ein Graph keine Dreiecksüberdeckung besitzt, exisitert kein valides + entsprechendes Zertifikat C∆ und der Verifizierer verwirft die Instanz. + Laufzeit: Es gibt maximal |V |3 verschiedene Teilmengen, die ein Dreieck bilden könnten und + daher überprüft werden müssen. Die Überprüfung geht in O(|V |) Zeit. Da alle anderen Schritte + des Verifizierers in O(1) Zeit erledigt werden können, ergibt sich somit insgesamt eine Laufzeit + von O(|V |4 ), was einer polynomiellen Laufzeit entspricht. + Also gilt ∆ C OVER ∈ NP. + Nun reduzieren wir V ERTEX C OVER auf ∆ C OVER. Betrachte folgende Reduktion: + Sei I = (G = (V, E), k) eine Instanz von V ERTEX C OVER. Bilde den Graphen G′ = (V ′ , E ′ ) mit + V ′ = V ∪ { ve | e ∈ E } und E ′ = E ∪ { {ve , e1 }, {ve , e2 } | e = {e1 , e2 } ∈ E }. Gebe I ′ = (G′ , k) + als ∆ C OVER Instanz aus. + Korrektheit: + ⇒ (Falls (G, k) eine Ja-Instanz von V ERTEX C OVER ist, dann ist (G′ , k) eine Ja-Instanz von ∆ + C OVER.): + Sei C ein Vertex Cover mit Größe kleiner oder gleich k von G. Dann ist C ebenfalls eine Drei- + ecksüberdeckung mit Größe kleiner oder gleich k. Offensichtlich gilt |C| ≤ k. Da C ein Vertex + Cover von G ist, gilt für jede Kante {u, v} ∈ E, dass u ∈ C oder v ∈ C. Für alle Teilgraphen + D = (VD , ED ) ⊆ (V, E) mit |VD | = 3 und ED = { {v1 , v2 } ∈ VD ×VD | v1 ̸= v2 } von G gilt damit + direkt, dass mindestens ein v ∈ VD auch in C enthalten ist. Der Graph G′ besitzt neben allen + solchen Teilgraphen in G zusätzlich solche Teilgraphen (Dreiecke), die durch die Knotenmenge + { ve | e ∈ E } und die entsprechenden Kanten { e, {ve , e1 }, {ve , e2 } | e = {e1 , e2 } ∈ E } entste- +hen. Für diese Dreiecke gilt aber auch, dass mindestens ein Knoten, nämlich e1 oder e2 , in C +enthalten sein muss, da C ein Vertex Cover ist. +⇐ (Falls (G′ , k) eine Ja-Instanz von ∆ C OVER ist, dann ist (G, k) eine Ja-Instanz von V ERTEX - +C OVER.): +Sei C∆ eine Dreiecksüberdeckung von G′ der Größe |C∆ | ≤ k. Ohne Beschränkung der All- +gemeinheit können wir davon ausgehen, dass C∆ ⊆ V gilt: Falls für einen Knoten ve ∈ C∆ gilt, +können wir e1 zu C∆ hinzufügen und ve aus C∆ löschen. Dadurch bleibt die Überdeckungseigen- +schaft erhalten und die Menge wird höchstens kleiner. Nun gilt also C∆ ⊆ V . Die Menge C∆ ist +auch ein Vertex Cover mit Größe kleiner oder gleich k von G. Offensichtlich gilt |C∆ | ≤ k. Da für +alle Teilgraphen D′ = (VD′ , ED′ ) ⊆ (V ′ , E ′ ) mit |VD′ | = 3 und ED′ = { {v1 , v2 } ∈ VD′ ×VD′ | v1 ̸= v2 } +von G′ gilt, dass mindestens ein Knoten v ∈ VD′ ebenfalls in C∆ enthalten sein muss, gilt dies +auch für die Dreiecke {ve , e1 , e2 }. Da außerdem C∆ ⊆ V gilt, gilt e1 ∈ C∆ oder e2 ∈ C∆ für alle +Kanten e ∈ E. +Laufzeit: Das Hinzufügen von einem neuen Knoten und zwei neuen Kanten pro Kante geht in +O(|E|). Damit geht die gesamte Reduktion in O(|E|), also in polynomieller Zeit. +Also existiert eine polynomielle Reduktion von V ERTEX C OVER auf ∆ C OVER. Da V ERTEX - +C OVER NP-vollständig aus den Präsenzaufgaben bekannt ist und ∆ C OVER ∈ NP gilt, ist somit +∆ C OVER NP-vollständig. + + +Bewertung (a) und (b) jeweils 1 Punkt, (c) gibt 8 Punkte: 1 Punkt Angabe Verifizierer/NDTM, +0,5 Punkte Korrektheit Verifizierer/NDTM, 0,5 Punkte Laufzeit Verifizierer/NDTM, 1 Punkt +Angabe Reduktion, 3 Punkte Korrektheit (je Richtung 1,5 Punkte), 1 Punkt Laufzeit Reduktion, +1 Punkt Beweisführung + + + + +Abgabe: 30.06.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 11 + Marek Lenczewski + Matrikelnummer: 1025252 + + 28. Juni 2025 + + +Hausaufgabe 11.1 +a + 1 + + + 2 3 + + + 4 5 + + + 6 10 + + + 7 9 + + + 8 + + +b + a b c + + + d e + f g + + h i j + + + k l m n + o p q + r s + + + t u v + + +c +Teil 1: ∆-Cover ∈ NP + + + 1 + Zuerst brauchen wir einen polynomiellen Verifizierer, um zu zeigen, dass ∆-Cover ∈ NP. +Der Verifizierer bekommt einen Graphen G = (V, E), Zahl k und ein Zertifikat C∆ ⊆ V und +arbeitet wie folgt: + + + 1. Prüfe, ob |C∆ | ≤ k + 2. Prüfe alle Knotentripel {a, b, c} ⊆ V , ob sie ein Dreieck bilden (also ob {a, b}, {b, c} und + {a, c} ∈ E zutrifft) und ob a, b oder c ∈ C∆ + 3. Ablehnen, falls ein Dreieck den test nicht besteht, sonst akzeptieren + + Die Schritte 1 und 3 laufen in O(1) und Schritt zwei läuft in O(|V |3 ), das ergibt eine poly- +nomielle Laufzeit. Damit ist ∆-Cover ∈ NP. + + Teil 2: ∆-Cover ist NP-schwer + Jetzt brauchen wir eine Reduktion von VertexCover auf ∆-Cover, um zu zeigen, dass +∆-Cover NP-schwer ist. + + Konstruktion: Sei dafür ein VertexCover gegeben mit einem Graphen G = (V, E) und +einer Zahl k. Aus G wird G′ = (V ′ , E ′ ), wobei für jede Kante e = {u, v} ∈ E ein neuer Kno- +ten we ∈ V ′ hinzugefügt wird und mit beiden Knoten u, v verbunden wird, sodass die Kanten +{v, we } und {u, we } zu E ′ dazu kommen. Das k = k ′ wird übernommen. So werden aus allen +Kanten Dreiecke. + + Beweis VertexCover nach ∆-Cover + • Sei C ein VertexCover für G mit |C| ≤ k + • Zu zeigen sei, dass C auch ein ∆-Cover für G′ mit |C| ≤ k ist + • Da C ein VertexCover ist, gilt u ∈ C oder v ∈ C + • Jedes Dreieck in G′ hat die Form {u, v, we } für eine Kante e = {u, v} ∈ E + • Somit hat jedes Dreieck mindestens einen Knoten in C + • Also ist C ein ∆-Cover für G′ + Beweis ∆-Cover nach VertexCover + • Sei C∆ ein ∆-Cover für G′ mit |C∆ | ≤ k ′ + • Jede Kante e = {u, v} ∈ E bildet durch {u, v, we } ein Dreieck in G′ + • Also gilt u, v oder we ∈ C∆ für alle Kanten in G′ + • Falls we ∈ C∆ , dann kann man we durch u oder v tauschen ohne die Eigenschaft von + ∆-Cover zu verletzen, sodass u oder v ∈ C∆ für alle Kanten in G′ gilt + • Da sich die Knoten aus C∆ nur vertauschen gilt weiter |C| = |C∆ | ≤ k + • Weiter sind in C∆ nur noch die Knoten aus V , somit gilt u ∈ C oder v ∈ C für alle Kanten + E + • Also ist C ein VertexCover für G + + + + Da ∆-Cover ∈ NP und NP-schwer ist, folgt dass ∆-Cover NP-vollständig ist. □ + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 30.06.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 12 + + +Hausaufgabe 12.1 (Lower Bound: V ERTEX C OVER) (5 Punkte) +Betrachten Sie die Reduktion aus der Musterlösung zu Präsenzserie 10 von C LIQUE auf V ER - +TEX C OVER . + + 1. Geben Sie die Anzahl der Knoten, die Anzahl der Kanten und den Wert k für die re- + sultierende V ERTEX C OVER Instanz in Abhängigkeit von der originalen C LIQUE Instanz + an. + 2. Zeigen Sie Lower Bounds für V ERTEX C OVER bezüglich Anzahl der Knoten, Anzahl + der Kanten und k basierend auf der ETH und der Reduktion aus der Musterlösung zu + Präsenzserie 10. + +Lösung. 1. Sei I = (G = (V, E), k) eine Instanz von C LIQUE und I ′ = (G′ = (V ′ , E ′ ), k′ ) die + aus der Reduktion resultierende Instanz von V ERTEX C OVER. Es gilt |V ′ | = |V |, |E ′ | = + 1 2 ′ + 2 |V |(|V | − 1) − |E| ≤ |V | und k = |V | − k ≤ |V |. + ′ + 2. • Angenommen es gäbe einen Algorithmus, der V ERTEX C OVER in 2o(|V |) · |I|O(1) + löst. Durch die Kombination der Reduktion von C LIQUE auf V ERTEX C OVER und + diesem Algorithmus, existiert dann auch ein 2o(|V |) ·|I|O(1) Algorithmus für C LIQUE, + da |V ′ | = |V | nach Aufgabenteil 1. Aus der Musterlösung der Präsenzserie 12 folgt + somit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für V ERTEX C OVER nur geben, wenn die ETH + falsch ist. + √ ′ + • Angenommen es gäbe einen Algorithmus, der V ERTEX C OVER in 2o( |E |) · |I|O(1) + löst. Durch die Kombination der Reduktion von C LIQUE auf V ERTEX C OVER und + diesem Algorithmus, existiert dann auch ein 2o(|V |) ·|I|O(1) Algorithmus für C LIQUE, + da |E ′ | ≤ |V |2 nach Aufgabenteil 1. Aus der Musterlösung der Präsenzserie 12 folgt + somit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für V ERTEX C OVER nur geben, wenn die ETH + falsch ist. + ′ + • Angenommen es gäbe einen Algorithmus, der V ERTEX C OVER in 2o(k ) · |I|O(1) löst. + Durch die Kombination der Reduktion von C LIQUE auf V ERTEX C OVER und die- + sem Algorithmus, existiert dann auch ein 2o(|V |) · |I|O(1) Algorithmus für C LIQUE, + da k′ ≤ |V | nach Aufgabenteil 1. Aus der Musterlösung der Präsenzserie 12 folgt + somit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für V ERTEX C OVER nur geben, wenn die ETH + falsch ist. + Bewertung 1 Punkt für den ersten Teil und 1.5 + 1.5 + 1 für den zweiten. + +Definition 1 (H ITTING S ET). Bei diesem Problem ist ein Universum U, Teilmengen des Uni- +versums F1 , . . . , Fr ⊆ U und eine Zahl k ∈ N≥0 gegeben. Gibt es eine Menge S ⊆ U mit |U| ≤ k +und S ∩ Fi ̸= 0/ für alle i ∈ [r]? +Anschaulicher wird bei diesem Problem nach maximal k Elementen aus dem Universum ge- +fragt, sodass diese Elemente alle gegebenen Mengen Fi „treffen“. + +Hausaufgabe 12.2 (Lower Bound: H ITTING S ET) (5 Punkte) +Betrachten Sie folgende Reduktion von 3-SAT auf H ITTING S ET. +Für eine gegebene 3-SAT Instanz I mit n Variablen x1 , . . . , xn und m Klauseln C1 , . . . ,Cm : Als +Universum wählen wir U = { x1 , x̄1 , . . . , xn , x̄n }. Für jede Variable xi fügen wir die Menge Fi := +{ xi , x̄i } hinzu. Für jede Klausel C j fügen wir die Menge Fn+ j := C j hinzu. Wir setzen k := n +und erhalten damit eine H ITTING S ET Instanz I. + 1. Geben Sie die Größe des Universums, die Anzahl der Mengen und die Zahl k aus der + resultierenden H ITTING S ET Instanz in Abhängigkeit von der Anzahl der Variablen n und + der Anzahl der Klauseln m aus der originalen 3-SAT Instanz an. + 2. Beweisen Sie Lower Bounds für H ITTING S ET bezüglich Universumsgröße, Anzahl der + Mengen und der Zahl k basierend auf der obigen Reduktion und der ETH. + +Lösung. 1. Für die Universumsgröße gilt |U| = 2n, für die Anzahl der Mengen gilt r = + n + m und es gilt k = n. + 2. • Angenommen es gäbe einen Algorithmus, der H ITTING S ET in 2o(|U|) · |I|O(1) löst. + Durch die Kombination der Reduktion von 3-SAT auf H ITTING S ET und diesem + Algorithmus, existiert dann auch ein 2o(n) · |I|O(1) Algorithmus für 3-SAT, da |U| = + 2n nach Aufgabenteil 1. Damit ist die ETH falsch. + Damit kann es so einen Algorithmus für H ITTING S ET nur geben, wenn die ETH + falsch ist. + • Angenommen es gäbe einen Algorithmus, der H ITTING S ET in 2o(r) · |I|O(1) löst. + Durch die Kombination der Reduktion von 3-SAT auf H ITTING S ET und diesem + Algorithmus, existiert dann auch ein 2o(m) · |I|O(1) Algorithmus für 3-SAT, da r = + n + m nach Aufgabenteil 1 und n ≤ 3m für 3-SAT gilt. Aus dem Sparsification + Lemma folgt damit, dass die ETH falsch ist. + Damit kann es so einen Algorithmus für H ITTING S ET nur geben, wenn die ETH + falsch ist. + • Angenommen es gäbe einen Algorithmus, der H ITTING S ET in 2o(k) · |I|O(1) löst. + Durch die Kombination der Reduktion von 3-SAT auf H ITTING S ET und diesem + Algorithmus, existiert dann auch ein 2o(n) · |I|O(1) Algorithmus für 3-SAT, da k = n + nach Aufgabenteil 1. Damit ist die ETH falsch. + Damit kann es so einen Algorithmus für H ITTING S ET nur geben, wenn die ETH + falsch ist. + + +Bewertung 1 Punkt für den ersten Teil und 1.5 + 1.5 + 1 für den zweiten. + +Abgabe: 07.07.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 12 + Marek Lenczewski + Matrikelnummer: 1025252 + + 6. Juli 2025 + + +Hausaufgabe 12.1.1 +Die Knotenzahl bleibt gleich: |V ′ | = |V | = n +Die Kantenzahl entspricht den nicht vorhandenen Kanten: |E ′ | = |E| +k ′ entspricht allen Knoten, die vorher nicht in der Clique waren: k ′ = n − k + + +Hausaufgabe 12.1.2 +Wir wissen, dass 3-SAT laut ETH mindestens 2Ω(n) braucht, wobei n für die Anzahl der Varia- +blen steht. + + In der Vorlesung wurde eine Reduktion von 3-SAT auf Clique gezeigt. In der Präsenzaufgabe +10.2 wurde eine Reduktion von Clique auf VertexCover gezeigt. Also gibt es eine Reduktion von +3-SAT zu VertexCover. Somit kann VertexCover nicht schneller lösbar sein als 3-SAT, weil sonst +3-SAT schneller werden würde, was nach ETH nicht geht. + + So ergeben sich die folgenden Lower Bounds: + + • Bezüglich Anzahl der Knoten: 2Ω(n) , wobei n = |V ′ |. Die Knotenzahl verändert sich + nicht bei den Reduktionen. + √ + • Bezüglich Anzahl der Kanten: 2Ω( m) , wobei m = |E ′ |. Die Kanten können bei den + Reduktionen zu einem quadratischen Wachstum der Kanten führen. + ′ + • Bezüglich k ′ : 2Ω(k ) . Bei der Reduktion gilt k ′ = n − k, wobei k ′ linear von n abhängt. + + + + + 1 + Hausaufgabe 12.2.1 + • Größe des Universums: |U | = 2n. Für jede der n Variablen gibt es zwei Elemente (xi + und xi ) + + • Anzahl der Mengen: n + m. n sind die Variablenmengen und m die Klauselmengen. (Fi + und Fn+j ) + + • Parameter: k = n. Wird in der Reduktion so übernommen. + + +Hausaufgabe 12.2.2 +Wir wissen, dass 3-SAT laut ETH mindestens 2Ω(n) braucht, wobei n für die Anzahl der Varia- +blen steht. Somit kann HittingSet nicht schneller sein. + + Es ergeben sich die folgenden Lower Bounds für HittingSet: + + • Bezüglich Größe des Universums: 2Ω(|U |/2) , da für jede Variable zwei Elemente ins + Universum aufgenommen werden + + • Bezüglich Anzahl der Mengen: 2Ω(n) , wobei die Anzahl der Variablen- und Klausel- + mengen (m + n) mindestens n ist + + • Bezüglich Zahl k: 2Ω(k) , da in der Reduktion k = n gilt. + + + + + 2 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 07.07.2025 + + Hausaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 13 + + +Definition 1 (M AX -3-S AT). Bei dem Problem M AX -3-S AT ist eine Formel φ in konjunktiver +Normalform gegeben, wobei jede Klausel 3 Literale enthält. Gesucht ist eine Belegung β der +Variablen, die die Anzahl v(β ) der erfüllten Klauseln maximiert. + +Hausaufgabe 13.1 (M AX -3-S AT) (10 Punkte) +Betrachten Sie folgenden Algorithmus A: + • Sei β0 die Belegung, die alle Variablen auf false setzt. + • Sei β1 die Belegung, die alle Variablen auf true setzt. + • Falls v(β0 ) ≥ v(β1 ), gib β0 zurück. + • Sonst gib β1 zurück. + + (a) (6 Punkte) Zeigen Sie, dass obiger Algorithmus Güte 2 hat, d.h. v(A(φ )) ≥ 12 v(OPT (φ )) + gilt für alle Eingaben φ . + (b) (4 Punkte) Geben Sie eine Formel an, bei der der Algorithmus eine Belegung ausgibt, die + genau die Hälfte der maximal erfüllbaren Klauseln erfüllt. Begründen Sie kurz, warum + Ihre Formel geeignet ist. + Hinweis: Sie brauchen nur höchstens zwei Klauseln und vier Variablen. + +Lösung. (a) Für jede Klausel C gilt, dass sie von mindestens einer der Belegungen β0 und + β1 erfüllt wird. Wir zeigen nun per Widerspruch, dass der obige Algorithmus eine Güte + von 2 hat: + Angenommen, es gilt v(A(φ )) < 12 v(OPT (φ )) für eine Formel φ . Sei m die Anzahl an + Klauseln in φ . Dann gilt v(OPT (φ )) ≤ m. Da v(A(φ )) < 12 v(OPT (φ )) ≤ 12 m, gilt v(β0 ) < + 1 1 + 2 m und v(β1 ) < 2 m. Da jede Klausel C allerdings von mindestens einer der Belegungen + β0 und β1 erfüllt wird, gilt v(β0 ) ≥ 12 m oder v(β1 ) ≥ 12 m. Dies ist ein Widerspruch. + Also gilt v(A(φ )) ≥ 21 v(OPT (φ )) für alle Eingaben φ und obiger Algorithmus hat Güte + 2. + (b) Betrachte folgende Formel: φ = (x1 ∨x2 ∨x3 )∧(¬x0 ∨¬x2 ∨¬x3 ) Mit β = {x0 → f alse, x1 → + true, x2 → true, x3 → true} gilt v(β ) = 2. Da φ nur zwei Klauseln hat, gilt somit v(OPT (φ )) = + 2. Es gilt allerdings v(β0 ) = 1, da hier nur die zweite Klausel erfüllt ist, und v(β1 ) = 1, + da hier nur die erste Klausel erfüllt ist. Somit gilt v(A(φ )) = 1 = 12 · 2 = 21 v(OPT (φ )). + Bewertung (a) 2 Punkte korrekte Beweisführung im Allgemeinen, 2 Punkte Abschätzung +Algorithmus, 2 Punkte Abschätzung Optimum; (b) 2 Punkte für Angabe einer korrekten Formel, +1 Punkt Ausgabe Algorithmus, 1 Punkt Optimum + + + + +Abgabe: 14.07.2025, 10:00 in Moodle – bitte beachten Sie die Informationen zum Abgabefor- +mat im Moodlekurs. + Hausaufgabe 13 + Marek Lenczewski + Matrikelnummer: 1025252 + + 13. Juli 2025 + + +Hausaufgabe 13.1.a +Zu zeigen ist, dass der Algorithmus die Güte 2 hat, also die Lösung vom Algorithmus mindestens +halb so gut ist wie die optimale Lösung. Also ν(A(ϕ)) ≥ 21 ν(OP T (ϕ)) für alle Eingaben ϕ. + + Sei ϕ eine beliebige 3-SAT-Formel mit m Klauseln C1 , C2 , . . . , Cm , wobei jede Klausel genau +3 Literale enthält. + + Jede Klausel Ci ist wahr, wenn β0 wahr ist oder wenn β1 wahr ist oder beides. Da die Varia- +blen in Ci nur positiv oder negativ sein können, muss dabei mindestens β0 oder β1 wahr werden. + + Der Algorithmus wählt die Belegung β0 oder β1 aus, die die meisten Klauseln Ci erfüllt. +Somit ist die Anzahl der ausgewählten Klauseln Ci mindestens m2 , da eine bei jeder Klausel +mindestens eine Belegung wahr werden muss und somit hat mindestens eine Belegung mindes- +tens m + 2 wahre Klauseln. + + Sei ν0 = Anzahl der von β0 erfüllten Klauseln und ν1 = Anzahl der von β1 erfüllten Klauseln. + + So ergibt sich ν(A(ϕ)) = max{ν0 , ν1 } ≥ ν0 +ν + 2 + 1 + ≥m + 2. + + Da die optimale Lösung höchstens alle m Klauseln erfüllen kann, gilt ν(OP T (ϕ)) ≤ m. + ν(OP T (ϕ)) + Daraus folgt: ν(A(ϕ)) ≥ m + 2 ≥ 2 . + + Somit hat der Algorithmus eine Güte von 2. + + + + + 1 + Hausaufgabe 13.1.b +Der Algorithmus erfüllt genau die Hälfte der maximal erfüllbaren Klauseln, wenn jede Klausel +nur aus positiven oder negativen Variablen besteht und sich diese genau gleich aufteilen. + + Formel: ϕ = (x1 ∨ x2 ∨ x3 ) ∧ (¬x1 ∨ ¬x2 ∨ ¬x3 ) + + Unter β0 (alle false): + + • C1 = (x1 ∨ x2 ∨ x3 ) = (false ∨ false ∨ false) = false + + • C2 = (¬x1 ∨ ¬x2 ∨ ¬x3 ) = (true ∨ true ∨ true) = true + + • ν(β0 ) = 1 + + Unter β1 (alle true): + + • C1 = (x1 ∨ x2 ∨ x3 ) = (true ∨ true ∨ true) = true + + • C2 = (¬x1 ∨ ¬x2 ∨ ¬x3 ) = (false ∨ false ∨ false) = false + + • ν(β1 ) = 1 + + Der Algorithmus wählt ν(A(ϕ)) = max{ν(β0 ), ν(β1 )} = max{1, 1} = 1. + + Eine optimale Lösung wäre zum Beispiel x1 = true, x2 = false, x3 = false. + + • C1 = (true ∨ false ∨ false) = true + + • C2 = (false ∨ true ∨ true) = true + + Daraus ergibt sich ν(OP T (ϕ)) = 2, also ν(A(ϕ)) = ν(OP2T (ϕ)) . 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Jansen + + 19. Juli 2024 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2024 + Name: Matrikel-Nr.: + + + Hinweise: + + • Bearbeiten Sie alle Aufgaben. + • Sie haben 10 Minuten Einlesezeit. Danach haben Sie 120 Minuten Zeit für die Klausurbearbeitung. + • Erlaubte Hilfsmittel: Ein doppelseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein + Rot, kein Grün, kein Bleistift). + • Einsichtnahme: 18.08.2024. + Note 4,0 3,7 3,3 3,0 2,7 2,3 2,0 1,7 1,3 1,0 + • + Mindestpunkzahl 17,5 19,5 21,5 23,5 25 27 29 31 33 35 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 12 + Aufgabe 1 KOMPLEXITÄT: A NWENDUNG LPT (4+2+2+2 Punkte) + Gegeben ist folgende Scheduling Instanz: + Jj J1 J2 J3 J4 J5 J6 J7 + m=3 + pj 5 3 7 8 1 4 2 + + a) Wenden Sie den LPT-Scheduling Algorithmus auf die obige Instanz an. Geben Sie dafür alle + relevanten Zwischenschritte an, sowie die Reihenfolge in der die Jobs auf Maschinen platziert + werden. Stellen Sie den resultierende Schedule graphisch dar. Eine beispielhafte Darstellung + ist: + + J3 + J6 + J2 + J5 J7 + J1 + J4 + m1 m2 m3 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 12 + b) Geben Sie die approximative Güte des Algorithmus LPT an. + + + + + c) Geben Sie eine Instanz für m = 2 Maschinen an, bei denen der Algorithmus ListScheduling + eine Güte von 2 − 1/m erzielt. Geben Sie zusätzlich an, welche Werte die optimale Lösung + OPT und die Lösung von LPT für diese Instanz erzielen. + + + + + d) Geben Sie eine Instanz für m = 3 Maschinen an, bei denen der Algorithmus ListScheduling + eine Güte von 2 − 1/m erzielt. Geben Sie zusätzlich an, welche Werte die optimale Lösung + OPT und die Lösung von LPT für diese Instanz erzielen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 12 + Aufgabe 2 KOMPLEXITÄT: B EWEIS : K -C LIQUE (10 Punkte) + Beweisen Sie folgende Aussage aus der Vorlesung: + SAT ≤ k-Clique. + Hinweis: NICHT 3-SAT, sondern allgemeines SAT! + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 12 + Aufgabe 3 KOMPLEXITÄT: NP-VOLLSTÄNDIGKEIT (5+5 Punkte) a) + Wir betrachten folgendes Problem: + Problem: CliqueAndIndependetSet + Eingabe: Ein ungerichteter Graph G = (V, E) mit V = 1, . . . , |V | und eine ganze Zahl k ≥ 1. + Entscheide: Gibt es in G sowohl ein Independent Set (Menge von Knoten, die paarweise + nicht adjazent sind, d.h. zwischen denen es keine Kanten gibt) der Größe k und als auch eine + Clique der Größe k? + Zeigen Sie die NP-Schwere von CliqueAndIndependetSet durch Angabe einer Reduktion + eines nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 12 + b) Wir betrachten folgendes Problem: + Problem: (a1 = 1)-SubsetSum + Eingabe: Eine Menge von n Items, jedes Item i ∈ [n] hat eine Größe ai ∈ N>0 , wobei a1 = 1, + und ein Zielwert T . + Entscheide: Gibt es I ⊆ [n] sodass ∑i∈I ai = T ? + Zeigen Sie die NP-Schwere von (a1 = 1)-SubsetSum durch Angabe einer Reduktion eines + nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 12 + Aufgabe 4 KOMPLEXITÄT: A PPROXIMATIVE A LGORITHMEN (3+7 Punkte) + Wir betrachten folgendes Problem: + Problem: A PPROXIMATE S UBSET S UM + Eingabe: n ganze Zahlen a1 , . . . an und ein Zielwert T . Es gilt ai ≤ T ∀i ∈ [n] sowie ∑i∈[n] ai > + T. + Ziel: Wähle Zahlenmenge S ⊆ {1, . . . n} mit ∑ j∈S a j ≤ T und maximiere ∑ j∈S a j . + + Betrachten Sie zu diesem Problem den folgenden Algorithmus GA. + + Algorithmus GA((A,T)) + + 1 sortiere Zahlen absteigend, sodass a1 ≥ a2 ≥ · · · ≥ an gilt; + 2 integer i = 1; + 3 integer sum = 0; + 4 boolean tooLarge = false; + 5 while i ≤ n and tooLarge = false do + 6 if sum+ai ≤ T then + 7 sum=sum+ai ; + 8 i = i + 1; + 9 else + 10 tooLarge = true; + 11 fi + 12 od + 13 return sum; + + a) Geben Sie eine Konstruktionsvorschrift für eine Instanz I an, mit der der Algorithmus GA + beliebig nah an eine Güte von 2 kommt, also A(I) ≈ OPT2 (I) . + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 12 + b) Zeigen Sie, dass der Algorithmus GA eine approximative Güte von 2 hat, also A(I) ≥ OPT2 (I) + gilt. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 12 + Aufgabe 5 KOMPLEXITÄT: ETH (5+5 Punkte) + + a) Betrachten Sie nachfolgende Reduktion von 3-SAT auf k-Color: + 1. Eliminiere doppelte Klauseln und Variablen, die in keiner Klausel auftauchen + 2. Elimiere Klauseln die eine Variable und ihre Negation enthalten (diese sind immer erfüllt) + 3. Seien C1 , . . . ,Cm die Klauseln in der SAT Formel + 4. Definiere  + V := { xi , x̄i , vi | i ∈ [n] } ∪ C j j ∈ [m] ∪ {z} , + sowie +  + E := {vi , v j }, {vi , x j }, {vi , x̄ j } i ∈ [n], j ∈ [n] \ {i} + ∪ { {xi , x̄i } | i ∈ [n] } +  + ∪ {xi ,C j } i ∈ [n], j ∈ [m], xi ∈ / Cj +  + ∪ {x̄i ,C j } i ∈ [n], j ∈ [m], x̄i ∈ / Cj + ∪ { {vi , z} | i ∈ [n] } +  + ∪ {C j , z} j ∈ [m] + Welche Lower Bounds ergeben sich unter Annahme der ETH for k-Colour aus dieser Re- + duktion in Hinblick auf + (i) die Anzahl der Knoten |V |; sowie + (ii) die Anzahl der Kanten |E|? + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 12 + b) Wir betrachten folgendes Scheduling Problem: + Problem: 2|prec, pi ∈ {1, 2}|Cmax + Eingabe: Eine Menge J von n Jobs, jeder Job J ∈ J hat eine Ausführungszeit pJ ∈ {1, 2} + sowie Präzedenzconstraints in Form eines gerichteten, azyklischem Graphen G = (J , A), + sowie ein ganze Zahl T ≥ 0. + Entscheide: Existiert ein Schedule, d.h. eine Abbildung, die jedem Job eine Startzeit und + eine Maschine zuweist, sodass (a) sich Jobs auf einer Maschine nicht überlappen (b) für alle + Jobs j, k ∈ J mit ( j, k) ∈ A, dass k erst startet nachdem j vollständig abgearbeitet ist und + (c) der Makespan höchstens T ist? + Wir betrachten die folgende Reduktion von k-Clique auf 2|prec, pi ∈ {1, 2}|Cmax . + Seien G = (V, E) und k gegeben. Wir setzen n = 4|V | + 3|E| und erstellen einen Knoten-Job Ji + für alle i ∈ V mit pi = 1. Des Weiteren erstellen wir Kanten-Jobs J{i, j} für jede Kante {i, j} ∈ E + mit p{i, j} = 2. Abschließend erstellen wir noch 3|V | + 2|E| Dummy Jobs. Diese Jobs haben + Ausführungszeit pdummy = 1. + Wir setzen unsere Präzedenzen nach dem folgenden Schema: + 1) (Ji , J{i, j} ) für alle i ∈ V, {i, j} ∈ E + 2) Präzedenzen innerhalb der Dummy Jobs. + Zuletzt setzen wir den Makespan T = 2|V | + 2|E|. + (i) Geben Sie die Lower Bounds für 2|prec, pi ∈ {1, 2}|Cmax in Hinblick auf n an, die sich + unter der ETH aus dieser Reduktion, basierend auf den Lower Bounds für k-Clique, + ergeben. + (ii) Geben Sie die Lower Bounds für 2|prec, pi ∈ {1, 2}|Cmax in Hinblick auf T an, die sich + unter der ETH aus dieser Reduktion, basierend auf den Lower Bounds für k-Clique, + ergeben. + Hinweis: Ein beispielhafter Schedule ist unten dargestellt. Die Präzedenzen sind im Graphen + durch gerichtete Kanten abgebildet, daher startet J4 erst nach dem Ende von J1 . Nutzen Sie + für diese Aufgabe die bekannten Lower Bounds aus der Präsenzübung. Diese wurden aus der + Reduktion von 3-SAT auf k-Clique hergeleitet. Hierbei ist der Clique Graph zusammenhän- + gend! + + J3 + + J1 + J3 J4 + J4 + + J1 J2 + J2 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 12 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 12 + Aufgabe 6 B ONUSAUFGABE (10 Punkte) + Beweisen oder widerlegen Sie folgende Aussage: + Wenn die ETH fehlschlägt gilt P = NP. + + + + + Viel Erfolg! + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 12 von 12 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen + + 11. Oktober 2024 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2024 + Name: Matrikel-Nr.: + Hinweise: + + • Bearbeiten Sie alle Aufgaben. + • Sie haben 10 Minuten Einlesezeit. Danach haben Sie 120 Minuten Zeit für die Klausurbearbeitung. + • Erlaubte Hilfsmittel: Ein doppelseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein + Rot, kein Grün, kein Bleistift). + • Einsichtnahme: 30.10.2024 + Note 4,0 3,7 3,3 3,0 2,7 2,3 2,0 1,7 1,3 1,0 + • + Mindestpunktzahl 17,5 19,5 21,5 23,5 25 27 29 31 33 35 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 16 + Aufgabe 1 ∆T SP1 (6+2+2 Punkte) + Gegeben sei folgender Graph G = (V, E), bei dem die Kantengewichte jeweils an der nach außen + gerichteten Seite der Kante stehen: + + a + 4 2 + + 3 + e b + 4 2 + + 4 3 + 6 5 + + d c + 2 + + a) Wenden Sie den Algorithmus ∆T SP1 auf den Graphen G beginnend bei a an. Geben Sie dabei + die Ergebnisse jedes Zwischenschrittes des Algorithmus an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 16 + b) Beschreiben Sie, was der Algorithmus ∆T SP1 berechnet und geben Sie die Bedingungen an, + die an den Graphen gestellt sind, damit der Algorithmus eine korrekte Lösung berechnet. + + + + + c) Begründen Sie, warum der Algorithmus ∆T SP1 eine approximative Güte von 2 besitzt. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 16 + Aufgabe 2 KOMPLEXITÄT: VORLESUNGSBEWEIS SAT (10 Punkte) + Beweisen Sie folgende Aussage: + 3-SAT ist NP-vollständig. Zeigen Sie dafür SAT ≤ 3-SAT, nehmen Sie also 3-SAT ∈ NP als + gegeben an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 16 + Aufgabe 3 KOMPLEXITÄT: NP-VOLLSTÄNDIGKEIT (5+5 Punkte) + + a) Wir betrachten folgendes Problem: + Problem: Hitchhiker’s-HamiltonianCycle + Eingabe: Ein ungerichteter Graph G = (V, E) mit V = {1, . . . , |V |} und für alle v ∈ V gilt + deg(v) ≥ 42. + Entscheide: Gibt es einen Hamiltonkreis in G? + Beweisen Sie die NP-Schwere von Hitchhiker’s-HamiltonianCycle durch Angabe einer + Reduktion eines nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + Hinweis: Als Hilfe, geben Sie graphisch die Reduktion für Knotengrad ≥ 10 an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 16 + b) Wir betrachten folgendes Problem: + Problem: AtMostTwoPerSize-SubsetSum + Eingabe: Ein ganzzahliger Zielwert T > 0, eine Menge von n Items, jedes Item i ∈ [n] + hat eine Größe ai ∈ N>0 , jede Größe tritt höchstens zwei Mal auf, d.h. für alle i ∈ [n] gilt + | { i′ ∈ [n] | ai′ = ai } | ≤ 2. + Entscheide: Gibt es S ⊆ [n] sodass ∑i∈S ai = T ? + Beweisen Sie die NP-Schwere von AtMostTwoPerSize-SubsetSum durch Angabe einer Re- + duktion eines nach Vorlesung/Übung bekanntermaßen NP-schweren Problems. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 16 + Aufgabe 4 KOMPLEXITÄT: A PPROXIMATIVE A LGORITHMEN (10+(10) Punkte) + Wir betrachten das aus der Vorlesung bekannte Makespan Scheduling Problem P||Cmax . Dazu ist + folgender Algorithmus gegeben: + + Algorithmus ROUND ROBIN S CHEDULING(I=(J,m)) + + 1 sortiere die Jobs in J so, dass p1 ≥ p2 ≥ · · · ≥ pn gilt; + 2 setze B1 = · · · = Bm = 0; + / + 3 integer j = 1; + 4 integer i = 1; + 5 while j <= n do + 6 Platziere Job J j auf Maschine Mi ; + 7 Bi = Bi ∪ {J j }; + 8 j = j + 1; + 9 if i < m then + 10 i=i+1 + 11 else + 12 i=1 + 13 fi + 14 od + 15 return B1 , . . . , Bm + + Betrachten Sie folgendes Beispiel zu der Funktionsweise des Round Robin Scheduling Algorith- + mus’. Gegeben sind 3 Maschinen und 6 Jobs J1 , . . . J6 , die bereits richtig sortiert sind. Es gilt + p1 = 7, p2 = 5, p3 = 5, p4 = 4, p5 = 3, p6 = 1. Daraus ergibt sich folgender Schedule: + + + + + J4 + + J5 + J6 + + J1 + J2 J3 + + + m1 m2 m3 + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 16 + a) Zeigen Sie per Widerspruch, dass der Algorithmus Round Robin Scheduling eine approxima- + tive Güte von 2 hat. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 16 + b) BONUS: Geben Sie die Konstruktionsvorschrift für eine Instanz an, mit der der Algorithmus + Round Robin Scheduling beliebig nah an eine Güte von 2 kommt, also A(I) ≈ 2 · OPT (I). + Nutzen Sie dafür mindestens 3 Maschinen und 7 Jobs, womit eine Rate von A(I) = 53 OPT + zu erreichen ist. Geben sie die optimale Makespan an und die, die der Algorithmus bei Ihrer + Instanz erreicht. Für eine Güte von 53 können bis zu 3 Punkten erreicht werden, bei einer + Güte von 47 sind bis zu 5 Punkte zu erreichen. Geben Sie eine Konstruktionsvorschrift an, + bei denen bei einer beliebigen, festen Anzahl von Maschinen m eine Güte von 2 − m1 erreicht + wird, können Sie die vollen 10 Punkte erreichen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 16 + Aufgabe 5 KOMPLEXITÄT: ETH (5+5 Punkte) + + a) Wir betrachten folgendes Problem: + Problem: DominatingSet + Eingabe: Ein ungerichteter Graph G = (V, E) mit V = 1, . . . , |V | und eine Ganzzahl k ≥ 1. + Entscheide: Gibt es eine Menge M ⊆ V der Kardinalität |M| = k derart, dass jeder Knoten + v ∈ V entweder in M ist oder zu einem Knoten in M benachbart ist. + Wir untersuchen nun folgende Reduktion von 3-SAT auf DominatingSet: + 1. Seien C1 , . . . ,Cm die Klauseln der 3-SAT-Formel + 2. Seien x1 , . . . , xn die Variablen der 3-SAT-Formel + 3. Erzeuge die Knoten xi , xi und di für jede Variable xi + 4. Erzeuge die Kanten {xi , xi }, {xi , di } und {di xi } für jede Variable xi + 5. Für jede Klausel C j erzeuge einen Knoten C j + 6. Für jedes Literal ℓ einer Klausel C j erzeuge die Kanten {C j , ℓ} + Welche unteren Schranken für die Laufzeit folgen damit für DominatingSet unter der ETH + in Bezug auf + (i) die Anzahl der Knoten |V |? + (ii) die Anzahl der Kanten |E|? + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 12 von 16 + b) Wir betrachten folgendes Problem: + Problem: GridTiling + Eingabe: Zwei ganze Zahlen k und j und für jedes i ∈ [k] und j ∈ [k] eine Menge Si, j ⊆ [n]2 . + Entscheide: Gibt es (ei, j )(i, j)∈[k]2 derart, dass + 1. für alle (i, j) ∈ [k]2 gilt ei, j ∈ Si, j ; + 2. für alle i ∈ [k] und j ∈ [k − 1] gilt für ei, j = (a, b) und ei, j+1 = (a′ , b′ ), dass a = a′ (dh. die + ausgewählten Tupel müssen in der ersten Komponente übereinstimmen, wenn der erste + Index derselbe ist); und + 3. für alle i ∈ [k − 1] und j ∈ [k] gilt für ei, j = (a, b) und ei+1, j = (a′ , b′ ), dass b = b′ (dh. + die ausgewählten Tupel müssen in der zweiten Komponente übereinstimmen, wenn der + zweite Index derselbe ist) + Betrachten Sie folgende Beispielinstanz für k = 2, die Elemente der Lösung sind unterstrichen: + + + S1,1 = {(1, 1), (2, 2)} S1,2 = {(1, 2), (2, 1), (2, 2)} + + + + S2,1 = {(1, 2), (2, 1)} S2,2 = {(1, 1), (2, 2)} + + + Wir untersuchen folgende Reduktion von Clique: + 1. Sei G = (V, E) und eine Zahl k als zu untersuchende Clique Instanz gegeben + 2. Wir nehmen ohne Beschränkung der Allgemeinheit an, dass V = {1, . . . , |V |} sowie k ≥ 2 + gilt und dass keine isolierten Knoten existieren. + 3. Wir übernehmen den Wert von k + 4. Für jedes Paar (i, j) ∈ [k]2 definieren wir + ( + { (a, a) | a ∈ V } if i = j + Si, j := + { (a, b) | a ̸= b, {a, b} ∈ E } if i ̸= j + Welche unteren Schranken für die Laufzeit folgen damit für Grid Tiling (mit der aus der + Vorlesung bekannten Reduktion von Clique auf 3-SAT) in Hinblick auf + (i) der Gesamtanzahl X := ∑(i, j)∈[k]2 |Si, j | der Tupel in den Mengen; sowie + (ii) der maximalen Anzahl Y := max(i, j)∈[k]2 |Si, j | an Tupeln in einer der Mengen? + Hinweis: Aus der Reduktion in der Vorlesung wissen wir, dass Clique selbst dann noch NP- + schwer ist, wenn k ∈ Θ(|V |) gilt. Sie dürfen also annehmen, dass es eine (globale) Konstante + c > 0 gibt, sodass c · n ≤ k ≤ n. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 13 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 14 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 15 von 16 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 16 von 16 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen + + 14. Juli 2023 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2023 + Name: Matrikel-Nr.: + + + Hinweise: + + • Bearbeiten Sie alle Aufgaben. + • Sie haben 10 Minuten Einlesezeit. Danach haben Sie 120 Minuten Zeit für die Klausurbearbeitung. + • Erlaubte Hilfsmittel: Ein doppelseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein + Rot, kein Grün, kein Bleistift). + • Einsichtnahme: Nach Absprache. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 11 + Aufgabe 1 RUCKSACKPROBLEM (3+2+5 Punkte) + Wir betrachten die Anwendung der Approximationsheuristiken für das Rucksackproblem aus der + Vorlesung. + Gegeben sei dafür folgende Instanz des Rucksackproblems mit Kapazität B = 16. Items sind in + der Form (pi , wi ) angegeben. + I = [(1, 4), (1, 1), (1, 3), (11, 13), (3, 3), (5, 6), (1, 5), 16] + a) Wenden Sie den normalen Greedy-Algorithmus auf obige Instanz an. Begründen Sie kurz, + warum die gewählten Items ausgewählt werden. Was ist der Lösungswert? + + + + + b) Wenden Sie nun den ModifiedGreedy-Algorithmus an. Wie ändert sich die Lösung? + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 11 + c) Was ist die Güte des ModifiedGreedy-Algorithmus? Geben Sie eine kurze Idee an, wie man + diesen Ansatz modifizieren kann, um die Güte auf 32 oder 43 zu verbessern. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 11 + Aufgabe 2 L ONGEST PATH (3+3+4 Punkte) + Betrachten Sie das folgende Problem: + Problem: L ONGEST PATH + Eingabe: Eine ungerichteter Graph G = (V, E) und eine Zahl k. + Entscheide: Gibt es einen Pfad in G, der mindestens Länge k hat? + + Zeigen Sie auf verschiedene Weisen, dass das Problem in NP liegt: + + a) Geben Sie ein Zertifikat für L ONGEST PATH an. + b) Beschreiben Sie einen Verifizierer für L ONGEST PATH und geben Sie dessen Laufzeit konkret + an. + c) Beschreiben Sie eine nicht-deterministische Turing-Maschine (in Worten), die L ONGEST PATH + löst. Geben Sie auch hier die Laufzeit konkret an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 11 + Aufgabe 3 A PPROXIMATIVE A LGORITHMEN (6+4 Punkte) + Das Problem M AX -3-S AT sei folgendermaßen definiert: + Problem: M AX -3-S AT + Eingabe: Eine Formel φ in konjunktiver Normalform, wobei jede Klausel drei Literale enthält. + Ausgabe: Eine Belegung β der Variablen, die die Anzahl v(β ) der erfüllten Klauseln maximiert. + + Betrachten Sie folgenden Algorithmus A: + + • Sei β0 die Belegung, die alle Variablen auf false setzt. + • Sei β1 die Belegung, die alle Variablen auf true setzt. + • Falls v(β0 ) ≥ v(β1 ), gib β0 zurück. + • Sonst gib β1 zurück. + + a) Zeigen Sie, dass obiger Algorithmus Güte 2 hat, d.h. A(φ ) ≥ 12 OPT (φ ) gilt für alle Eingaben + φ. + b) Geben Sie eine Formel an, bei der der Algorithmus eine Belegung ausgibt, die genau die Hälfte + der maximal erfüllbaren Klauseln erfüllt. Begründen Sie kurz, warum Ihre Formel geeignet ist. + Hinweis: Sie brauchen nur höchstens zwei Klauseln und sechs Variablen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 11 + Aufgabe 4 B EWEIS ZUR VORLESUNG (10 Punkte) + Betrachten Sie folgendes Problem aus der Vorlesung: + + Problem: k-C LIQUE + Eingabe: Ein ungerichteter Graph G = (V, E) und eine Zahl k ≥ 1. Eine Clique ist eine Teil- + menge C ⊆ V mit {u, v} ∈ E für alle u, v ∈ C mit u ̸= v. + Entscheide: Hat der gegebene Graph G eine Clique C ⊆ V mit mindestens k Knoten (d.h. mit + |C| ≥ k)? + + Beweisen Sie: Das Problem k-C LIQUE ist NP-vollständig. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 11 + Aufgabe 5 NP UND NP-VOLLSTÄNDIGKEIT (4+4+7 Punkte) + + + a) Beweisen Sie die NP-Vollständigkeit von S UBSET S UM, bei der jede Itemgröße jeweils durch + 3 oder durch 7 teilbar ist. + b) Beweisen Sie die NP-Vollständigkeit von 3-C OLOR, wobei jeder Knoten im Graph mindes- + tens Grad 3 hat. + c) Zeigen Sie: Für kein α > 1 gibt es einen approximativen Algorithmus mit Güte α (d.h. mit + Zielfunktionswert ≤ α · OPT) für TSP, außer P = NP (Denken Sie an Wiliam Rowan Hamil- + ton). + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 11 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 11 + Aufgabe 6 U NTERE S CHRANKEN UND ETH (7+8 Punkte) + Betrachten Sie folgende Reduktionen: + Problem: H ITTING S ET + Eingabe: Eine Menge U sowie r Teilmengen F1 , . . . , Fr ⊆ U und eine Zahl k. + Entscheide: Gibt es eine Menge S ⊆ U mit |S| ≤ k und S ∩ Fi ̸= 0/ für alle i ∈ [r]? + + a) 3-SAT ⪯ H ITTING S ET: Seien v1 , . . . , vn die Variablen der Formel und C1 , . . . ,Cm die Klau- + seln. Setze U = { v1 , v1 , . . . , vn , vn } und k = n. Für jede Variable vi definiere eine Menge + Fi = {vi , vi } und für jede Klausel C j definiere eine Menge Fn+ j = C j + Welche unteren Schranken ergeben sich durch obige Reduktion für die Laufzeit von Algorith- + men für H ITTING S ET unter Annahme der ETH in Abhängigkeit von + (i) r? + (ii) |U|? + Beweisen Sie die entsprechenden unteren Schranken! + Hinweis: Sie dürfen – wie in den Hausaufgaben zur ETH – Faktoren, die polynomiell in der + Eingabekodierung sind, vernachlässigen. + b) 3-S AT ⪯ S UBSET S UM: Seien v1 , . . . , vn die Variablen der Formel und C1 , . . . ,Cm die Klauseln. + Für jede Variable vi erzeuge zwei Items ai und bi , mit den Größen + + s(ai ) = 10i−1 + ∑ 10n+ j−1 und s(bi ) = 10i−1 + ∑ 10n+ j−1 . + j∈[m] j∈[m] + xi ∈C j xi ∈C j + + Zusätzlich erzeuge zwei Items c j und d j für jede Klausel C j mit Größen + + s(c j ) = s(d j ) = 10n+ j−1 . + + Der Zielwert sei + B = ∑ 3 · 10n+ j−1 + ∑ 10i−1 . +  + j∈[m] i∈[n] + + Welche unteren Schranken ergeben sich durch obige Reduktion für die Laufzeit von Algorith- + men für S UBSET S UM unter Annahme der ETH in Abhängigkeit von + (i) der Anzahl der unterschiedlichen Itemgrößen d? + (ii) der binären Kodierungslänge L der größten Itemgröße ∆ (d.h. L = log(∆))? + Beweisen Sie die entsprechenden unteren Schranken! + Hinweis: Sie dürfen auch hier – wie in den Hausaufgaben zur ETH – Faktoren, die polynomiell + in der Eingabekodierung sind, vernachlässigen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 11 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 11 + Viel Erfolg! + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 11 + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen + + 19. Oktober 2023 + + Modulprüfung zur Vorlesung + »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« + SS 2023 + Name: Matrikel-Nr.: + + +Hinweise: + +• Bearbeiten Sie alle Aufgaben. +• Sie haben 120 Minuten Zeit für die Klausurbearbeitung. +• Erlaubte Hilfsmittel: Ein einseitig handschriftlich beschriebenes Blatt Papier, (bunte) Stifte (kein Rot, + kein Grün, kein Bleistift, kein Tintenkiller, kein Tipp-Ex). +• Schreiben Sie auf jedes Blatt ihren Namen. +• Wenn Sie Schmierzettel verwenden um eine Aufgabe zu bearbeiten, verweisen Sie in der Aufgabe auf + diese. +• Einsichtnahme: 2.11.23, 13:30-14:30 Uhr. +• Die Klausur hat 6 Aufgaben auf 13 Seiten – überprüfen Sie bitte vor der Bearbeitung, dass Ihre Klausur + vollständig ist. + Aufgabe 1 C HRISTOFIDES ’ A LGORITHMUS (3+7 Punkte) + Gegeben sei folgender Graph G = (V, E) : + + d + + 2 + 3 + + 3 + a c + + + 1 + 2 + + b + + a) Geben Sie an, was Christofides’ Algorithmus berechnet, und welche approximative Güte er + hat. Welche Eigenschaft müssen die Distanzen neben der Symmetrie noch erfüllen, damit + Christofides’ Algorithmus korrekt arbeitet. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 1 von 19 + b) Wenden Sie den Algorithmus von Christofides auf den Graphen G mit Startknoten a an. Ge- + ben Sie dabei alle Graphen an, die in Zwischenschritten entstehen. Erwähnen Sie ebenfalls + Schritte, die bei dieser Anwendung zu keiner Änderung führen, die der Algorithmus aber + überprüfen muss. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 2 von 19 + Aufgabe 2 D OMINATING S ET (3+3+4 Punkte) + Betrachten Sie das folgende Problem: + Problem: D OMINATING S ET + Eingabe: Ein ungerichteter Graph G = (V, E) (wobei V = {1, . . . , n} für eine Zahl n ∈ N≥1 ) + und eine Zahl k ∈ N0 . + Entscheide: Gibt es eine Teilmenge D ⊆ V der Knoten mit |D| ≤ k, sodass jeder Knoten v ∈ V + in D enthalten ist oder benachbart zu mindestens einem der Knoten in D ist? + + Zeigen Sie auf verschiedene Weisen, dass das Problem in NP liegt: + a) Geben Sie ein Zertifikat für D OMINATING S ET an. + b) Beschreiben Sie einen polynomiellen Verifizierer für D OMINATING S ET und schätzen Sie die + Laufzeit in O-Notation konkret ab. + c) Beschreiben Sie (in Worten) eine polynomielle, nicht-deterministische Turing-Maschine, die + D OMINATING S ET löst. Schätzen Sie auch hier die Laufzeit in O-Notation konkret ab. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 3 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 4 von 19 + Aufgabe 3 A PPROXIMATIVE A LGORITHMEN (6+4 Punkte) + Das Problem M IN -E DGE -C OVER sei folgendermaßen definiert: + Problem: M IN -E DGE -C OVER + Eingabe: Ein zusammenhängender, ungerichteter Graph G = (V, E). + Ausgabe: Eine Teilmenge der Kanten C ⊆ E mit minimaler Kardinalität, sodass jeder Knoten + v ∈ V zu mindestens einer Kante aus C inzident ist. + + Betrachten Sie folgenden Algorithmus A: + + • Gehe (in beliebiger Reihenfolge) alle Knoten durch: + • Falls der aktuelle Knoten v noch nicht abgedeckt ist, wähle zufällig eine der zu v inzidenten + Kanten und füge sie zu C hinzu. + • Gib anschließend die entstehende Menge C zurück. + + a) Zeigen Sie, dass obiger Algorithmus Güte 2 hat, d.h. A(G) ≤ 2 · OPT(G) gilt für alle Eingaben + G. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 5 von 19 + b) Sei n ∈ N≥4 gerade. Betrachten Sie den Graphen Gn , der genau n Knoten und n Kanten hat und + einen Kreis beschreibt. D.h. Gn = ({1, . . . , n}, {{i, i+1} | i ∈ {1, . . . , n−1}}∪{{n, 1}}). Wie viele + Kanten hat ein M IN -E DGE -C OVER des Graphen Gn ? Wie schlecht ist die Approximationsrate + des Algorithmus A für Gn im schlimmsten Fall? Geben Sie eine Reihenfolge der Knoten (für + Schritt 1) und eine Auswahl der Kanten (für Schritt 2) an, die zu einer möglichst schlechten + Approximationsrate führen. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 6 von 19 + Aufgabe 4 B EWEIS – S CHEDULING AUF IDENTISCHEN M ASCHINEN (10 Punkte) + Betrachten Sie das Scheduling-Problem P||Cmax : + Problem: P||Cmax + Eingabe: Eine Liste L = (J1 , . . . , Jn ) von n Jobs mit Ausführungszeiten p1 , . . . , pn ∈ N, und m + identische Maschinen. + Ausgabe: Ein Schedule (Partition von J in m Teilmengen B1 , . . . , Bm ) mit minimaler maximaler + Last Cmax := max ∑ p j . + 1≤i≤m J ∈B + j i + + + Betrachten Sie den folgenden Algorithmus ListScheduling, der einen Schedule berechnet. Sei + LS(I) die Last Cmax eines Schedules des Algorithmus, und OPT(I) die Last Cmax eines optimalen + Schedules zur Eingabe I. Zeigen Sie: Für alle Eingaben I = (L, m) gilt LS(I)/OPT(I) ≤ 2 − 1/m. + + Algorithmus L IST S CHEDULING(L = (J1 , . . . , Jn ), m) + + 1 for i = 1 to m do + 2 Ei = 0; Bi = 0/ ; + 3 od + 4 for j = 1 to n do + 5 wähle Job J j aus Liste L; + 6 wähle Maschine Mi mit minimaler Last Ei ; + 7 Bi = Bi ∪ {J j }; + 8 Ei = Ei + p j ; + 9 od + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 7 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 8 von 19 + Aufgabe 5 NP UND NP-VOLLSTÄNDIGKEIT (4+5+6 Punkte) + + Problem: H AMILTONIAN PATH + Eingabe: Ein ungerichteter Graph G = (V, E) + Entscheide: Existiert ein Pfad in G, der jeden Knoten genau einmal besucht? + Problem: H AMILTONIAN C YCLE + Eingabe: Ein ungerichteter Graph G = (V, E) + Entscheide: Existiert ein Kreis in G, der jeden Knoten genau einmal besucht? + + (a) Sei L ⊆ S UBSET S UM die Menge der (positiven) SubsetSum-Instanzen, bei denen keine der + Itemgrößen eine Zweierpotenz ist. Zeigen Sie, dass L NP-vollständig ist. + (b) Geben Sie eine polynomielle Reduktion von H AMILTONIAN PATH auf H AMILTONIAN C Y- + CLE an. + + (c) Geben Sie eine polynomielle Reduktion von H AMILTONIAN C YCLE auf H AMILTONIAN - + PATH an. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 9 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 10 von 19 + Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 11 von 19 + Aufgabe 6 U NTERE S CHRANKEN UND ETH (5+5 Punkte) + Im Folgenden bezeichne ⟨I⟩ die Kodierungslänge der Instanz. + (a) Wir betrachten folgendes Problem: + Problem: H ITTING S ET + Eingabe: Eine Menge U ′ sowie r′ Teilmengen F1′ , . . . , Fr′′ ⊆ U ′ und eine Zahl k′ . + Entscheide: Gibt es eine Menge S′ ⊆ U ′ mit |S′ | ≤ k′ und S′ ∩ Fi′ ̸= 0/ für alle i′ ∈ {1, . . . , r′ }? + + Sie dürfen nutzen, dass sich H ITTING S ET unter Annahme der ETH nicht in Zeit 2o(r ) ⟨I⟩O(1) + ′ + + + und nicht in Zeit 2o(|U |) ⟨I⟩O(1) lösen lässt. + ′ + + + Problem: S ET C OVER + Eingabe: Eine Menge U sowie r Teilmengen F1 , . . . , Fr ⊆ U und eine Zahl k. + Entscheide: Gibt es eine Menge S ⊆ {1, . . . , r} mit |S| ≤ k und i∈S Fi = U? + S + + + Welche unteren Schranken ergeben sich + (i) in Hinblick auf |U| sowie + (ii) in Hinblick auf r + für S ET C OVER unter Annahme der ETH aus den unteren Schranken für H ITTING S ET, wenn + man folgende Reduktion von SetCover auf HittingSet verwendet? + Sei (F ,U, k) eine S ET C OVER-Instanz. Setze V1 := U und V2 := {1, . . . , r}. Sei V := V1 ∪V˙ 2. + Weiter sei E = { {v, w} | v ∈ V1 , w ∈ V2 , v ∈ Fw }. Damit ist G = (V, E) ein bipartiter Graph + mit Partitionen V1 ,V2 . Setze U ′ := {1, . . . , r}. Erzeuge nun eine Menge Fw′ für jeden Knoten + w ∈ V1 , indem Fw′ := { v ∈ V2 | {v, w} ∈ E } gesetzt wird. Sei F ′ := { Fw′ | w ∈ V1 }. Mit k′ = k + iie berechnete Instanz ist nun durch (F ′ ,U ′ , k′ ) gegeben. + + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 12 von 19 + (b) Wir wollen uns mit einem Problem beschäftigen, das ein mächtiges Werkzeug im Bereich + der Optimierung ist: Integrale Lineare Programme (ILPs). + Problem: ILP-F EASIBILITY + Eingabe: Eine Matrix A ∈ ZM×N , eine rechte Seite b ∈ ZM . + Entscheide: Gibt es einen Vektor x ∈ ZN≥0 mit Ax ≤ b, d.h. Ai xi ≤ bi für alle i ∈ {1, . . . , M}? + Welche unteren Schranken ergeben sich + (i) in Hinblick auf M sowie + (ii) im Hinblick auf N + für ILP-F EASIBILITY unter Annahme der ETH, wenn man folgende Reduktion von 3-SAT + auf ILP-F EASIBILITY verwendet? + Sei φ eine 3-SAT Instanz mit m Klauseln (Ci )i∈m und n Variablen. Analog zu linearen Glei- + chungssystemen (Ax = b), die aus M Gleichungen bestehen, haben wir bei Ax ≤ b einen + Satz von M linearen Ungleichungen. Wir konstruieren diese wie folgt: Wir nutzen N := 2n + Variablen in unserem ILP, eine für jedes (mögliche) Literal in der SAT-Formel. Für je- + de SAT-Variable v sei xv die ILP-Variable, die zum SAT-Literal v korrespondiert, und xv̄ + die, die zum SAT-Literal v̄ korrespondiert. Für jede Klausel Ci erzeugen wir eine Unglei- + chung − ∑ℓ∈Ci xℓ ≤ −1. Weiter erzeugen wir für jede SAT-Variable v zwei Ungleichungen: + xv +xv̄ ≤ 1 und −xv −xv̄ ≤ −1. Durch das Eintragen der entsprechenden Koeffizienten in eine + erweiterte Koeffizientenmatrix (bestehend aus Matrix A und rechter Seite b) erhalten wir die + ILP-F EASIBILITY Instanz. + + + + + Viel Erfolg! + + + +Klausur »Analyse von Algorithmen und Komplexität (Komplexitätsteil)« Seite 13 von 19 + \ No newline at end of file diff --git a/uni/aak/alle_praesenzaufgaben.pdf b/uni/aak/alle_praesenzaufgaben.pdf new file mode 100644 index 0000000000000000000000000000000000000000..73af44c15d9b60655408c6d4af3b25a70d43fef2 GIT binary patch literal 648498 zcmd421z43^w>B&xEl7jZqC@HK?(XjHTBLM$BOu)k(j|&?NOvh+(yi3DP;`5a@7X85 z{hj~6F0aM2o;Bkh_q@lLF{Y$)LL$`kG>mYh?=Nz0;phQ$09ykKI4&*#t+X*v-$);* z4`2knQ~=P*>6;om0+@b$QF6002G9zEezvkT1@Q2|8QU1$S7i9D0e&Z-nXLnWf)Z5F z!B`(?Zfhf?4>Sf)2yrmbF)-7y(lOC9G0`!yJ*T512mSk5?!JzczLK#Tz{MPB1~3D4 z<=|j!0{U6n*62TLLC(R}$jQ(c)P$Xp38)Q_aZGeH3@nttfxI{9rs z{cBnx!1p3ROKu`Nb4z0oej*+?0GET235SA^$U`mAH*O*`AkdD3me$3^g~o-E#@4}< 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0 HcmV?d00001 diff --git a/uni/aak/alle_praesenzaufgaben.txt b/uni/aak/alle_praesenzaufgaben.txt new file mode 100644 index 0000000..6f985bd --- /dev/null +++ b/uni/aak/alle_praesenzaufgaben.txt @@ -0,0 +1,480 @@ + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 09.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 9 + + +Definition 1 (Knapsack (Entscheidungsvariante)). Gegeben ist eine Menge mit n Gegenständen +mit Gewichten w1 , . . . , wn ∈ N≥0 und Profiten p1 , . . . , pn ∈ N≥0 . Weiter sind eine Kapazität +K ∈ N≥0 und ein Zielprofit P ∈ N≥0 gegeben. Bei dem K NAPSACK Problem muss entschieden +werden, ob eine Teilmenge S ⊆ [n] existiert, sodass + + ∑ wi ≤ K und ∑ pi ≥ P + i∈S i∈S + +gilt. + +Präsenzaufgabe 9.1 (K NAPSACK in NP) +Zeigen Sie K NAPSACK ∈ NP auf zwei verschiedene Wege. + 1. Geben Sie einen nicht deterministischen Algorithmus an, der das Problem in polynomi- + eller Zeit löst. + 2. Geben Sie einen polynomiellen Verifizierer für das Problem an. +Wie hängen diese beiden Methoden zusammen? + +Lösung. 1. Der Algorithmus arbeitet wie folgt: Für jeden Gegenstand entscheiden wir nicht- + deterministisch, ob er in den Rucksack soll. Anschließend akzeptieren wir, falls die Kapazitäts- + und Profitbedingung erfüllt sind. + Existiert eine valide Rucksackfüllung mit genügend Profit, dann gibt es auch eine Reihe + von nichtdeterministischen Entscheidungen, die diese Füllung findet. Diese wird dann + auch vom Algorithmus akzeptiert. + Der Algorithmus macht polynomiell viele nichtdeterministische Entscheidungen und die + Überprüfung der Eigenschaften ist ebenfalls in Polynomialzeit möglich, also ist dies ein + polynomieller, nichtdeterministischer Algorithmus, der K NAPSACK löst. + 2. Unser Zertifikat ist eine Teilmenge S ⊆ [n], die die Auswahl für den Rucksack darstellen + soll. Dies kann zum Beispiel über ein Bit pro Gegenstand realisiert werden. Um dieses + Zertifikat zu verifizieren, berechnen wir ∑i∈S wi und ∑i∈S pi und vergleichen die Werte + mit K und P und akzeptieren entsprechend. + Existiert eine valide Rucksackfüllung mit genügend Profit, dann gibt es auch ein Zertifi- + kat, das diese Füllung beschreibt. Dieses wird dann auch vom Verifizierer akzeptiert. + Das Ausrechnen der Summen und das Überprüfen ist in polynomieller Zeit möglich, also + ist dies ein polynomieller Verifizierer. + In beiden Wegen wird die extra Ressource, die wir zur Verfügung haben, genutzt, um eine +Teilmenge (einen möglichen Rucksack) zu bestimmen und zu überprüfen. In der Vorlesung +wird noch genau gezeigt, dass diese beiden Wege tatsächlich äquivalent sind. + +Präsenzaufgabe 9.2 (Transitivität von Polynomialzeitreduktionen) +Zeigen Sie, dass Polynomialzeitreduktionen transitiv sind, d.h., dass für alle Entscheidungspro- +bleme L1 , L2 , L3 ⊆ Σ∗ gilt, dass + + L1 ⪯ L2 ∧ L2 ⪯ L3 =⇒ L1 ⪯ L3 . + +Lösung. Seien L1 , L2 , L3 Sprachen. Zudem sei R1 ein Algorithmus, der L1 nach L2 in O(na ) +reduziert, und R2 ein Algorithmus, der L2 nach L3 in O(nb ) reduziert. +Sei x eine Instanz von L1 der Größe n. Dann benötigt der Aufruf von R1 auf x höchstens O(na ) +Zeit. +Sei x′ die Ausgabe von R1 . Dann hat x′ eine Größe von höchstens O(na ). Diese Schranke für +die Ausgabe ergibt sich aus der Laufzeit der Reduktion. Der Aufruf von R2 auf x′ benötigt nun +höchstens O((na )b ) = O(nab ) Zeit. Damit ist die Reduktion von L1 auf L3 in polynomieller Zeit +möglich. +Die Korrektheit ergibt sich aus der Transitivität der Äquivalenz: Sei x′′ die Ausgabe von R2 auf +x′ , dann gilt + x ∈ L1 ⇐⇒ x′ ∈ L2 und x′ ∈ L2 ⇐⇒ x′′ ∈ L3 , +also auch x ∈ L1 ⇐⇒ x′′ ∈ L3 . + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 16.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 10 + + +Präsenzaufgabe 10.1 (Fragen über Fragen) +Diskutieren Sie die folgenden Aussagen (Gelten sie allgemein? Gelten sie nicht? Gelten sie +unter bestimmten Bedingungen? Welche?): + (i) Für jede Sprache, die von einer NDTM in polynomieller Zeit akzeptiert werden kann, + existiert eine DTM, die ebenfalls die Sprache in polynomieller Zeit akzeptiert. + (ii) A ist NP-schwer =⇒ A ∈ P + +Lösung. (i) Die Aussage gilt nur, wenn P = NP. Dann liegen alle Probleme in NP (von + NDTM in polynomieller Zeit lösbar) auch in P (von DTM in polynomieller Zeit lösbar). + Falls P ̸= NP, dann nein, weil sonst folgender Widerspruch auftritt: ein Problem in NP \ P + ist von einer NDTM, nicht aber von einer DTM in polynomieller Zeit lösbar. Könnte eine + DTM die NDTM in polynomieller Zeit simulieren, wäre das Problem auch in P. + (ii) Falsch. Es gibt NP-schwere Probleme, die nicht in NP liegen, z.B. das H ALTE-Problem. + Diese liegen folglich auch definitiv nicht in P. NP-schwer ist eine untere Schranke, ∈ NP + eine obere, und wenn beide erfüllt sind, ist ein Problem NP-vollständig. + + + +Definition 1 (Vertex Cover (VC)). Für einen (ungerichteten) Graphen G = (V, E) ist ein Vertex +Cover eine Menge von Knoten C ⊆ V , sodass u ∈ C oder v ∈ C für alle { u, v } ∈ E gilt, also ist +für alle Kanten mindestens ein Endpunkt in C. +Für das V ERTEX C OVER Problem ist neben einem Graphen G eine Zahl k ∈ N≥0 gegeben und +es wird gefragt, ob ein Vertex Cover mit Größe (Anzahl der Knoten) höchstens k in G existiert. + +Präsenzaufgabe 10.2 (V ERTEX C OVER ist NP-vollständig) +Zeigen Sie, dass V ERTEX C OVER NP-vollständig ist. +Hinweis: Reduzieren Sie C LIQUE auf V ERTEX C OVER. + +Lösung. V ERTEX C OVER ∈ NP ist bekannt aus der letzten Hausaufgabe. +Vorüberlegung zur Reduktion von C LIQUE auf V ERTEX C OVER: Sei eine Instanz mit G = +(V, E) und k ∈ N≥0 von C LIQUE gegeben. Wir invertieren zunächst den Graphen G und erhal- +ten damit G′ . Nun enthält G′ genau dann eine unabhängige Menge der Größe ℓ, wenn G eine +Clique der Größe ℓ enthält. Wenn ein Graph mit n Knoten eine unabhängige Menge mit min- +destens k Knoten hat, dann hat der Graph auch ein Vertex Cover mit maximal n − k Knoten + (das Komplement zur unabhängigen Menge). Unsere Instanz für V ERTEX C OVER ist also G′ +und n − k. (k ≤ n können wir annehmen) +Betrachte nun folgende Reduktion: +Sei eine Instanz mit G = (V, E) und k ∈ N≥0 von C LIQUE gegeben. Invertiere den Graphen G +und erhalte damit G′ . Gebe nun G′ und n − k als Instanz von V ERTEX C OVER zurück. +Korrektheit: +⇒ (Falls (G, k) eine Ja-Instanz von C LIQUE ist, dann ist (G′ , n − k) eine Ja-Instanz von V ER - +TEX C OVER .): + +G enthalte eine Clique C der Größe ≥ k. Dann ist V ′ := V \C ein Vertex Cover von G′ der Größe +≤ n − k. Da |C| ≥ k, gilt |V ′ | ≤ n − k. Da C eine Clique ist, also alle Knoten aus C mit allen +anderen Knoten aus C verbunden sind, gibt es im invertierten Graphen G′ keine Kante zwischen +Knoten aus C. Also gilt für jede Kante {u, v} ∈ E ′ , dass u ∈ / C und somit u ∈ V ′ oder + / C oder v ∈ +v ∈ V ′. +⇐ (Falls (G′ , n − k) eine Ja-Instanz von V ERTEX C OVER ist, dann ist (G, k) eine Ja-Instanz von +C LIQUE.): +G′ enthalte ein Vertex Cover V ′ der Größe ≤ n−k. Dann ist C := V \V ′ eine Clique der Größe ≥ +k in G. Da |V ′ | ≤ n − k, gilt |C| ≥ k. Für jede Menge {u, v} ⊆ C gilt, dass {u, v} ∈ + / E ′ . Ansonsten +wäre V ′ kein korrektes Vertex Cover. Da G′ der invertierte Graph von G ist, gilt somit {u, v} ∈ E +für jede Menge {u, v} ⊆ C. +Laufzeit: Das Invertieren des Graphen kann naiv in O(|V |2 · |E|) Zeit, also in polynomieller +Zeit, erledigt werden. +Also existiert eine polynomielle Reduktion von C LIQUE auf V ERTEX C OVER. Da C LIQUE +NP-vollständig ist und V ERTEX C OVER ∈ NP gilt, ist somit V ERTEX C OVER NP-vollständig. + + +Präsenzaufgabe 10.3 (Turingmaschinen) + n +Entwerfen Sie eine Turingmaschine für die Sprache L = { 02 | n ∈ N } über dem Alphabet +Σ = {0} und geben Sie die Laufzeit Ihrer Turingmaschine an. Begründen Sie die Korrektheit +und die Laufzeit Ihrer Turingmaschine. Diskutieren Sie die Laufzeit Ihrer Turingmaschine im +Vergleich zu einem Algorithmus (mit RAM) für diese Sprache. + +Lösung. Die Turingmaschine sieht folgendermaßen aus: + (1) Falls genau eine 0 auf dem Band steht, akzeptiere. + (2) Bewege den Kopf von links nach rechts über die Eingabe und ersetze jede zweite 0 durch + ein neues Symbol x. + (3) Falls das letzte Symbol der Eingabe nun eine 0 ist, verwerfe. + (4) Gehe zu Schritt (1). +Korrektheit: Die Sprache L besteht aus allen Wörtern, die nur Nullen enthalten und deren Länge +eine Zweierpotenz ist. Eine Zahl ist genau dann eine Zweierpotenz, wenn sie nach wiederholtem +Teilen durch 2 irgendwann eine 1 ergibt. Genau das überprüft die Turingmaschine. Wenn bei +diesem wiederholten Teilen eine ungerade Zahl auftritt, ist die Zahl keine Zweierpotenz. In +diesem Fall verwirft die Turingmaschine die Eingabe (siehe Schritt (3)). +Laufzeit: Die angegebene Turingmaschine hat eine Laufzeit von O(n log(n)), wobei n hier die +Länge der Eingabe bezeichnet. Schritt (1) und Schritt (2) benötigen jeweils eine Laufzeit von + O(n). Schritt (3) geht in konstanter Laufzeit O(1). Schritt (4) wiederholt die Schritte (1) bis (3) +bis nur noch eine 0 auf dem Band steht oder vorher abgebrochen wird. Da in jedem Durchlauf +der Schritte (1) bis (3) die Hälfte der Nullen (abgerundet) durch x Symbole ersetzt wird, steht +nach log n Durchläufen nur noch eine Null auf dem Band und es wird nach Schritt (1) akzeptiert +oder es steht nach ⌊log n⌋ − 1 Durchläufen eine ungerade Anzahl von Nullen auf dem Band und +es wird nach Schritt (3) verworfen. Also gibt es maximal log n + 1 Durchläufe. +Diskussion: Ein Algorithmus mit RAM könnte folgendermaßen aussehen: + 1. Zähle die Buchstaben und erhalte Anzahl m. + 2. Falls log m ganzzahlig ist, akzeptiere. Sonst verwerfe. +Dieser Algorithmus hat eine Laufzeit von O(n), wobei n hier die Länge der Eingabe bezeichnet. +Ein Algorithmus mit RAM kann also den zusätzlichen Speicher nutzen, um wesentlich schneller +eine Aufgabe zu erledigen, im Vergleich zu einem Algorithmus, der auf einer Turingmaschine +läuft. + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 23.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 11 + + +Präsenzaufgabe 11.1 (Fragen über Fragen) +Diskutieren Sie nachfolgende Aussagen. Sind diese wahr oder falsch? +(a) Für L ∈ NP gilt, aus L ⪯ 3-SAT folgt, dass L NP-vollständig ist. +(b) Gilt für eine Sprache L, dass sowohl L ⪯ 3-SAT als auch 3-SAT ⪯ L, dann ist L NP-vollständig. +(c) Sei L NP-vollständig. Dann gilt L ∈ P genau dann, wenn P = NP. +(d) Es ist möglich, dass 3-SAT ∈ P und C LIQUE ∈ + / P. + +Lösung. + / Offensichtlich gilt L ∈ NP. Aus L ⪯ 3-SAT folgt nur, +(a) Nein, ein Gegenbeispiel ist L = 0. + dass sich L auf 3-SAT reduzieren lässt, nicht anders herum. +(b) Das ist die Definition von NP-vollständig. +(c) Korrekt, da sich jedes Problem auf L reduzieren lässt nach Def. NP-vollständig (und offen- + sichtlich P ⊆ NP). +(d) Nein, weil 3-SAT NP-vollständig ist, lässt sich C LIQUE (liegt bekannterweise in NP, ist + sogar NP-vollständig) auf 3-SAT reduzieren. + + + +Präsenzaufgabe 11.2 (NP-Vollständigkeit) +In dem Problem C LIQUE -N OMEMBER ist ein Graph G = (V, E), ein Knoten v ∈ V sowie eine +Zahl k ∈ N gegeben und es soll entschieden werden, ob es eine Clique mit k Knoten gibt, die +den Knoten v nicht enthält. +Zeigen Sie, dass C LIQUE -N OMEMBER NP-vollständig ist, indem Sie eine Reduktion von C LI - +QUE auf C LIQUE -N OMEMBER angeben. Sie können davon ausgehen, dass C LIQUE -N OMEMBER ∈ +NP ist. + +Lösung. Clique-Nomember ist NP-schwer: +Sei (G = (V, E), k) eine Eingabe für das Clique Problem. +Idee: Füge einen neuen isolierten Knoten v zu G hinzu. +Wir geben zuerst die Abbildung an: Sei v ̸∈ V . Wir definieren die Eingabe für das Problem +C LIQUE -N OMEMBER als den Graphen G′ := (V ∪ {v}, E), den Knoten v als Knoten, der nicht + benutzt werden darf, und k als gesuchte Cliquengröße. (G′ , v, k) ist eine gültige Eingabe für +C LIQUE -N OMEMBER und in polynomieller Zeit berechenbar. +Wir zeigen nun: G hat eine k-Clique genau dann, wenn G′ eine k-Clique ohne v beinhaltet. +⇒: +Sei (G, k) eine Ja-Instanz des C LIQUE Problems. Dann enthält G eine k-Clique C. Dann ist C +auch eine k-Clique in G′ , da bei der Abbildung keine Kanten entfernt wurden. Da v ̸∈ V , enthält +C den Knoten v nicht. Damit ist C eine k-Clique in G ohne den Knoten v. (G′ , v, k) ist also eine +Ja-Instanz des C LIQUE -N OMEMBER Problems. +⇐: +Sei (G′ , v, k) eine Ja-Instanz des C LIQUE -N OMEMBER Problems. Dann enthält G′ eine k-Clique +C mit v ̸∈ C. Da bei der Abbildung an dem Graphen nichts geändert wurde, außer der Knoten v +hinzugefügt, muss C auch eine k-Clique in G sein. Damit ist (G, k) eine Ja-Instanz des C LIQUE +Problems. +Also existiert eine polynomielle Reduktion von C LIQUE auf C LIQUE -N OMEMBER. Da C LI - +QUE NP-vollständig ist und C LIQUE -N OMEMBER ∈ NP nach Aufgabenstellung gilt, ist somit +C LIQUE -N OMEMBER NP-vollständig. + + + +Präsenzaufgabe 11.3 (NP-Vollständigkeit) +Für das Problem S UBSET S UM C ARDINALITY seien n ganze Zahlen c1 , . . . , cn ∈ N>0 (wobei n +gerade sei) sowie eine Zahl K ∈ N gegeben. Es soll entschieden werden, ob es eine Teilmenge +S ⊂ {1, . . . , n} gibt mit |S| = n/2 und ∑i∈S ci = K. +Zeigen Sie, dass das Problem S UBSET S UM C ARDINALITY NP-vollständig ist. Sie können +davon ausgehen, dass S UBSET S UM C ARDINALITY ∈ NP ist. + +Lösung. Subset Sum Cardinality ist NP-schwer: +S UBSET S UM Problem: Gegeben: n ganze Zahlen c1 , . . . , cn ∈ N>0 und Zahl K ∈ N. Frage: Gibt +es eine Teilmenge S ⊆ {0, . . . , n} mit ∑i∈S ci = K? +Reduktion: Sei I = (c1 , . . . , cn , K) eine Eingabe für das S UBSET S UM Problem. Definiere c′i := +ci + 1 für i ≤ n und c′i := 1 für i ∈ {n + 1, . . . , 2n}. Wir definieren die Eingabe für das Problem +S UBSET S UM C ARDINALITY als I ′ = (c′1 , . . . , c′2n , K + n). +Wir zeigen nun: Es gibt eine Lösung für I genau dann, wenn es eine Lösung für I ′ gibt. +⇒: +Sei I eine Ja-Instanz des S UBSET S UM Problems. Dann gibt es eine Menge S ⊆ {1, . . . , n} mit +∑i∈S ci = K. Es gilt also ∑i∈S c′i = ∑i∈S (ci + 1) = K + |S|. Sei nun S′ = {n + 1, . . . , 2n − |S|}. Es +gilt |S ∪ S′ | = n und ∑i∈S∪S′ c′i = ∑i∈S c′i + ∑i∈S′ c′i = ∑i∈S (ci + 1) + ∑i∈S′ 1 = K + |S| + (n − |S|) = +K + n. S ∪ S′ ist also eine Lösung für I ′ und I ′ damit eine Ja-Instanz. +⇐: +Es sei I ′ eine Ja-Instanz. Dann gibt es also eine Menge S′ ⊆ {1, . . . , 2n} mit ∑i∈S′ c′i = K + n und +|S′ | = n. Sei S := S′ ∩{1, . . . , n}. Es gilt ∑i∈S′ c′i − ∑i∈S c′i = n−|S|, da die Items in {n+1, . . . , 2n} +das Gewicht 1 haben, und genau n − |S| Items übrig sind. Damit gilt ∑i∈S (ci + 1) = ∑i∈S c′i = +K + n − (n − |S|) = K + |S|, woraus folgt, dass ∑i∈S ci = K. Damit ist S eine Lösung für I und +somit I eine Ja-Instanz. + Laufzeit: Die Reduktion kann naiv in einer Zeit von O(n) implementiert werden. Das ist eine +polynomielle Laufzeit. +Also existiert eine polynomielle Reduktion von S UBSET S UM auf S UBSET S UM C ARDINALI - +TY . Da nach Skript S UBSET S UM NP-vollständig ist und S UBSET S UM C ARDINALITY ∈ NP +nach Aufgabenstellung gilt, ist somit S UBSET S UM C ARDINALITY NP-vollständig. + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 30.06.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 12 + + +Präsenzaufgabe 12.1 +Im Folgenden sollen Sie herausfinden (und begründen), welche Lower Bounds sich über ETH +und die Reduktionen aus der Vorlesung ergeben. Nutzen Sie hierfür auch das Sparsification +Lemma. +Welche Schranken für die Laufzeit in Abhängigkeit von den natürlichen Parametern (Anzahl +Knoten, Anzahl Kanten, Anzahl Items, Anzahl unterschiedlicher Items, Anzahl Tripel, Kardi- +nalität der Grundmenge, Anzahl unterschiedlicher Tripel, Anzahl unterschiedlicher auftretender +Zahlen, größte auftretende Zahl, . . . ) ergeben sich jeweils aus folgenden (zum Großteil aus der +Vorlesung bekannten) Reduktionen (Laufzeitfaktoren, die polynomiell in der Eingabekodierung +sind, dürfen vernachlässigt werden)? +Schätzen Sie dazu zunächst die verschiedenen Parameter der resultierenden Instanz mithilfe der +Parameter der eingegebenen Instanz geeignet ab. + 1. 3-SAT ⪯ k-C LIQUE + 2. k-C LIQUE ⪯ k-I NDEPENDENT S ET (Problemdefinition für die Antwort nicht relevant; in + der Reduktion wird der Graph G = (V, E) auf den Komplementgraph (V, E) abgebildet) + 3. SAT ⪯ 3-D IMENSIONAL M ATCHING (Hinweis: 3-SAT ⊆ SAT) + 4. 3-D IMENSIONAL M ATCHING ⪯ 3-E XACT C OVER + 5. 3-E XACT C OVER ⪯ S UBSET S UM + +Lösung. Angenommen ETH gilt. Das Sparsification Lemma impliziert, dass man 3-SAT nicht +in 2o(m) · |I|O(1) lösen kann. + + 1. In der Reduktion werden |V | = O(m) Knoten und |E| = O(m2 ) Kanten erstellt und k = m. + Außerdem gilt n ≤ 3m ∈ O(m). + • Lower bound bzgl. der Anzahl an Knoten: 2o(|V |) · |I|O(1) + Angenommen, es existiert ein Algorithmus, der k-Clique in 2o(|V |) ·|I|O(1) löst. Dann + gibt es auch einen Algorithmus, der 3-SAT in 2o(m) · |I|O(1) löst (Reduktion auf k- + Clique + k-Cliquen Algorithmus). Nach dem Sparsification Lemma gibt es dies nur, + wenn die ETH nicht gilt. + √ + • Lower bound bzgl. der Anzahl an Kanten: 2o( |E|) · |I|O(1) √ + Angenommen, es existiert ein Algorithmus, der k-Clique in 2o( |E|) · |I|O(1) löst. + Da |E| ∈ O(m2 ), gibt es dann auch einen Algorithmus, der 3-SAT in 2o(m) · |I|O(1) + löst (Reduktion auf k-Clique + k-Cliquen Algorithmus). Nach dem Sparsification + Lemma gibt es dies nur, wenn die ETH nicht gilt. + • Lower bound bzgl. k: 2o(k) · |I|O(1) + Angenommen, es existiert ein Algorithmus, der k-Clique in 2o(k) · |I|O(1) löst. Dann + gibt es auch einen Algorithmus, der 3-SAT in 2o(m) · |I|O(1) löst (Reduktion auf k- + Clique + k-Cliquen Algorithmus). Nach dem Sparsification Lemma gibt es dies nur, + wenn die ETH nicht gilt. +2. In der Reduktion werden genau |V | Knoten und maximal |E| = O(|V |2 ) Kanten erstellt. + • Lower bound bzgl. der Anzahl an Knoten: 2o(|V |) · |I|O(1) + Angenommen, es existiert ein Algorithmus, der k-Independent-Set in 2o(|V |) · |I|O(1) + löst. Dann gibt es auch einen Algorithmus, der k-Clique in 2o(|V |) · |I|O(1) löst (Re- + duktion auf k-Independent-Set + k-Independent-Set Algorithmus). Nach Aufgaben- + teil 1, gibt es diesen nur, wenn die ETH nicht gilt. + √ + o( |E|) + • Lower bound bzgl. der Anzahl an Kanten: 2 · |I|O(1) √ + Angenommen, es existiert ein Algorithmus, der k-Independen-Set in 2o( |E|) ·|I|O(1) + löst. Da |E| ≤ |V |2 , gibt es dann auch einen Algorithmus, der k-Clique in 2o(|V |) · + |I|O(1) löst (Reduktion auf k-Independent-Set + k-Independent-Set Algorithmus). + Nach Aufgabenteil 1, gibt es diesen nur, wenn die ETH nicht gilt. +3. In der Reduktion werden Mengen mit |U| = |V | = |W | = O(mn) Elementen erstellt und + |T | = O(m2 n2 ) Tupel. Es gilt n ≤ 3m ∈ O(m) für 3-SAT. + √ + • Lower bound bzgl. der Anzahl an Elementen: 2o( |V |) · |I|O(1) √ + Angenommen, es existiert ein Algorithmus, der 3-Dimensional-Matching in 2o( |V |) · + |I|O(1) löst. Da |V | ∈ O(mn) ≤ O(m2 ), gibt es dann auch einen Algorithmus, der + 3-SAT in 2o(m) · |I|O(1) löst (Reduktion von SAT auf 3-Dimensional-Matching an- + gewandt auf 3-SAT Instanzen + 3-Dimensional-Matching Algorithmus). Nach dem + Sparsification Lemma geht dies nur, wenn die ETH nicht gilt. + √ + 4 + • Lower bound bzgl. der Kardinalität von T : 2o( |T |) · |I|O(1) √ + 4 + Angenommen, es existiert ein Algorithmus, der 3-Dimensional-Matching in 2o( |T |) · + |I|O(1) löst. Da |T | ∈ O(m2 n2 ) ≤ O(m4 ), gibt es dann auch einen Algorithmus, der + 3-SAT in 2o(m) · |I|O(1) löst (Reduktion von SAT auf 3-Dimensional-Matching an- + gewandt auf 3-SAT Instanzen + 3-Dimensional-Matching Algorithmus). Nach dem + Sparsification Lemma geht dies nur, wenn die ETH nicht gilt. +4. Das 3-dimensionale Matching ist ein Spezialfall vom 3-Exact Cover. Daher gelten die + Lower Bounds von 3-dimensionalem Matching auch für 3-Exact Cover. Da |U| = 3m ∈ + O(|V |), ergibt sich, dass es für 3-Exact Cover unter der ETH keinen Algorithmus √ mit + √ + o( m) O(1) o( |V |) + Laufzeit 2 · |I| geben kann, da sonst 3-dimensionales Matching in 2 · + |I|O(1) lösbar wäre. Und da |F| = |T |, √ ergibt sich, dass es für 3-Exact Cover unter der ETH + o( 4 |F|) + keinen Algorithmus + √ mit Laufzeit 2 · |I|O(1) geben kann, da sonst 3-dimensionales + 4 + Matching in 2o( |T |) · |I|O(1) lösbar wäre. + √ + 4 +5. In der Reduktion wird n = |F| gesetzt. Lower bound bzgl. n: 2o( n)√· |I|O(1) + 4 + Angenommen, es existiert ein Algorithmus, der SubsetSum in 2o( n) · |I|O(1) löst. Dann + √ + 4 + |F|) +gibt es auch einen Algorithmus, der 3-Exact Cover in 2o( · |I|O(1) löst. Nach Aufga- +benteil 4, gibt es diesen nur, wenn die ETH nicht gilt. + CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL + Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität + Prof. Dr. K. Jansen, B. Schumacher, C. Wambsganz + + 07.07.2025 + + Präsenzaufgaben zur Vorlesung »Analyse von Algorithmen und + Komplexität« + Blatt 13 + + +Präsenzaufgabe 13.1 (Vertex Cover) +Ein Vertex Cover in einem Graphen G = (V, E) ist eine Menge von Knoten C ⊆ V , so dass für +jede Kante {v, w} ∈ E gilt, dass v ∈ C oder w ∈ C. Beim Optimierungsproblem Minimum Vertex +Cover soll die kardinalitätskleinste Menge C ⊆ V gefunden werden, die ein Vertex Cover von +G ist. +Gegeben ist der folgende approximative Algorithmus mit Güte 2 für VertexCover: +Idee: Wähle eine beliebige Kante. Wenn einer der beiden Endknoten bereits in C ist, verwirf +diese Kante. Sonst, füge beide Endknoten zur Menge C hinzu. Mache dies für jede Kante. + +Algorithmus 2A PPROX CV(G = (V, E)) + + 1 C = {} + 2 foreach {u, v} ∈ E do + 3 if u ̸∈ C and v ̸∈ C then + 4 C = C ∪ {u, v} + 5 fi + 6 od + 7 return C + +Beweisen Sie die Güte. + +Lösung. Beweis: Sei C∗ ein minimales Vertex Cover. Wir zeigen |C| ≤ 2|C∗ |. +Sei A die Menge der Kanten, für die beide Endknoten zu C hinzugefügt wurden. Es gilt |C| = +2|A|, da beide Knoten jeder ausgewählten Kante zu C hinzugefügt wurden und alle anderen +Kanten nicht weiter betrachtet werden. Damit folgt auch, dass keine zwei Kanten aus A einen +Knoten gemeinsam haben können. Sei nun C∗ ein minimales Vertex Cover. C∗ enthält nach +Definition mindestens einen Knoten jeder Kante, also insbesondere einen Knoten jeder Kante +aus A. Da keine zwei Kanten aus A vom gleichen Knoten aus C∗ überdeckt werden können, gilt +|C∗ | ≥ |A|. Es folgt |C| = 2|A| ≤ 2|C∗ |. Die Lösung C von 2ApproxCV ist also maximal doppelt +so groß wie die optimale Lösung C∗ . Damit berechnet der Algorithmus eine 2-Approximation. + Präsenzaufgabe 13.2 (TSP) +In der Vorlesung haben Sie zwei approximative Algorithmen für das TSP-Problem kennen ge- +lernt. Der erste Algorithmus ∆T SP1 hat eine Approximationsrate von 2, während der zweite +Algorithmus ∆T SP2 von Christofides eine Approximationsrate von 3/2 besitzt. In dieser Übung +soll gezeigt werden, dass die zweite Rate scharf ist. D.h., dass es eine Folge von Instanzen gibt, +auf denen der Algorithmus immer näher an diese Rate heran kommt. +Wir betrachten den vollständigen Graphen Kn = (V, E) für ein n ∈ N≥6 mit n mod 4 = 2 und +die nachfolgende Gewichtsfunktion d2 : E → R+ . Zeigen Sie, dass + (a) OPT((Kn , d2 )) = n gilt und + (b) eine mögliche Ausführung von ∆TSP2 existiert, die auf Eingabe (Kn , d2 ) eine Tour der + Länge (n − 1) + n/2 berechnet. + + + 1 1 1 1 + 1 3 5 7 9 n−1 + + + 1 1 1 1 1 1 + + 1 1 1 1 + 2 4 6 8 10 n + + + +Wir definieren E2 := {{i, i + 2}|i ∈ {1, . . . , n − 2}} ∪ {{2i + 1, 2i + 2}|i ∈ {0, . . . , n/2 − 1}} und +d2 , so dass alle Kanten aus E2 auf 1 abgebildet werden und alle Kanten aus E \ E2 auf den Wert +des kürzesten Wegs zwischen diesen Knoten in dem oben abgebildeten Graphen abgebildet +werden. + +Lösung. Wichtig bei dieser Aufgabe ist der folgende Fakt: Bei beiden Algorithmen wird ir- +gendein MST genommen und dann irgendeine Reihenfolge der Knoten. Typischerweise wird +diese Reihenfolge durch bestimmte Knoten- oder Kanten-Labels festgelegt (wird zum Beispiel +der Kruskal-Algorithmus zur Konstruktion des MST genutzt, so werden dort die Kanten nach +Gewicht sortiert. Kanten gleichen Gewichts müssen also auch noch durch irgendeine Reihen- +folge festgelegt werden.) Wir gehen also davon aus, dass wir diese Reihenfolge auch wählen +können. Sollte der Algorithmus also einen “besseren” MST oder eine “bessere” Knotenreihen- +folge nutzen, passen wir die Labels dementsprechend an. +Ein möglicher MST ist durch die unten angegebene Zick-Zack-Tour gegeben oder formal durch + h [ i + ET = {{ j, j + 1}, { j + 1, j + 3}, { j + 3, j + 2}, { j + 2, j + 4}}} ∪ {{n − 1, n}}. + 1≤ j≤n−5 + j mod 4=1 + + + 1 1 1 1 + 1 3 5 7 n−3 n−1 + + + 1 1 1 1 1 1 + + 1 1 1 1 + 2 4 6 8 n−2 n + Es gilt d(T ) = n − 1 für die Kosten d(T ) des MST T , da nur Kanten mit Kosten 1 gewählt +werden und n − 1 Kanten gewählt werden müssen, um den Zusammenhang sicher zu stellen. +Da jeder Knoten im MST außer 1 und n geraden Grad hat, ist das Matching K sehr simpel, +nämlich K = {{1, n}}. Die Kosten der Kante {1, n} sind n/2 (der kürzeste Pfad von 1 zu n) +und somit gilt d(K) = n/2 für die Kosten von K. Die Rundreise R entspricht einfach dem MST +vereinigt mit der Matchingkante und hat somit Kosten: d(R) = n − 1 + n/2. +Die optimale Tour würde erst “oben” und dann “unten” entlang laufen, also von Knoten 1 zu +Knoten 3 gehen, dann zu Knoten 5 und so weiter, bis Knoten n − 1 erreicht wurde. Danach +folgen Knoten n, n − 2, n − 4 und so weiter. Diese Tour hat offensichtlich Kosten n, da wir n +Kanten mit Gewicht 1 nutzen. Somit gilt OPT((Kn , d2 )) = n und wir haben eine Rate von + + ∆T SP2 ((Kn , d2 )) n − 1 + n/2 1 + = = 3/2 − ≤ 3/2. + OPT((Kn , d2 )) n n + +Da limn→∞ 3/2 − n1 = 3/2, kommen wir der Rate von 3/2 also beliebig nahe. + \ No newline at end of file

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