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lernmaterial/aufgaben.aux
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193
lernmaterial/aufgaben.fdb_latexmk
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lernmaterial/aufgaben.fdb_latexmk
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(rewritten before read)
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307
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72
lernmaterial/aufgaben.toc
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\contentsline {section}{\numberline {1}Anwendung}{2}{}%
|
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|
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|
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|
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|
||||
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|
||||
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|
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|
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|
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|
||||
\contentsline {subsection}{26\ \ NP-Vollständigkeit zeigen für 3-COLOR mit Minimalgrad 3}{34}{}%
|
||||
\contentsline {subsection}{27\ \ NP-Vollständigkeit zeigen für HamiltonianPath}{36}{}%
|
||||
\contentsline {subsection}{28\ \ NP-Vollständigkeit zeigen für Hitchhiker's-HamiltonianCycle}{38}{}%
|
||||
\contentsline {subsection}{29\ \ NP-Vollständigkeit zeigen für TSP-Entscheidung}{40}{}%
|
||||
\contentsline {subsection}{30\ \ NP-Vollständigkeit zeigen für $k$-CLIQUE-DEG-3}{41}{}%
|
||||
\contentsline {subsection}{31\ \ NP-Vollständigkeit zeigen für Partition}{43}{}%
|
||||
\contentsline {subsection}{32\ \ NP-Vollständigkeit zeigen für SubsetSumCardinality}{44}{}%
|
||||
\contentsline {subsection}{33\ \ NP-Vollständigkeit zeigen für $(a_1{=}1)$-SubsetSum}{45}{}%
|
||||
\contentsline {subsection}{34\ \ NP-Vollständigkeit zeigen für SubsetSum mit Teilbarkeit}{46}{}%
|
||||
\contentsline {subsection}{35\ \ NP-Vollständigkeit zeigen für SubsetSum ohne Zweierpotenzen}{47}{}%
|
||||
\contentsline {subsection}{36\ \ NP-Vollständigkeit zeigen für AtMostTwoPerSize-SubsetSum}{48}{}%
|
||||
\contentsline {subsection}{37\ \ NP-Schwere zeigen für HALT$_{\text {TM}}$}{50}{}%
|
||||
\contentsline {section}{\numberline {4}Approximative Algorithmen}{51}{}%
|
||||
\contentsline {subsection}{38\ \ Güte zeigen für ListScheduling}{52}{}%
|
||||
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|
||||
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|
||||
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|
||||
\contentsline {subsection}{42\ \ Güte zeigen für ModifiedGreedy}{56}{}%
|
||||
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|
||||
\contentsline {subsection}{44\ \ Güte widerlegen für Greedy}{58}{}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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|
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|
||||
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|
||||
\contentsline {subsection}{55\ \ ETH-Schranke zeigen für 3-DM}{70}{}%
|
||||
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|
||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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\contentsline {subsection}{67\ \ ETH-Schranke zeigen für ILP-Feasibility}{82}{}%
|
||||
77
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251
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lernmaterial/aufgabenkatalog.toc
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|
||||
\contentsline {section}{\numberline {1}NP-Zugehörigkeit}{2}{section.1}%
|
||||
\contentsline {subsection}{A1\ \ Longest Path $\in $ NP (Dreischritt)}{2}{section*.2}%
|
||||
\contentsline {subsection}{A2\ \ Dominating Set $\in $ NP (Dreischritt)}{4}{section*.3}%
|
||||
\contentsline {section}{\numberline {2}Vorlesungsbeweise}{6}{section.2}%
|
||||
\contentsline {subsection}{A3\ \ $P \subseteq NP$}{6}{section*.4}%
|
||||
\contentsline {subsection}{A4\ \ NDTM-Definition $\subseteq $ Verifizierer-Definition}{7}{section*.5}%
|
||||
\contentsline {subsection}{A5\ \ Verifizierer-Definition $\subseteq $ NDTM-Definition}{9}{section*.6}%
|
||||
\contentsline {subsection}{A6\ \ Transitivität von Polynomialzeitreduktionen}{10}{section*.7}%
|
||||
\contentsline {subsection}{A7\ \ Vererbung der NP-Vollständigkeit}{11}{section*.8}%
|
||||
\contentsline {subsection}{A8\ \ $L_0$ NP-vollständig und $L_0 \in P$ $\Rightarrow $ $P = NP$}{12}{section*.9}%
|
||||
\contentsline {subsection}{A9\ \ SAT $\leq _{p}$ 3-SAT}{13}{section*.10}%
|
||||
\contentsline {subsection}{A10\ \ $k$-Clique ist NP-vollständig}{15}{section*.11}%
|
||||
\contentsline {subsection}{A11\ \ SubsetSum $\leq _{p}$ Knapsack}{17}{section*.12}%
|
||||
\contentsline {subsection}{A12\ \ SubsetSum $\leq _{p}$ Partition}{18}{section*.13}%
|
||||
\contentsline {subsection}{A13\ \ ListScheduling hat Güte $2 - \frac {1}{m}$}{20}{section*.14}%
|
||||
\contentsline {subsection}{A14\ \ ModifiedGreedy hat Güte 2}{22}{section*.15}%
|
||||
\contentsline {section}{\numberline {3}Reduktionen: NP-Schwere und NP-Vollständigkeit}{23}{section.3}%
|
||||
\contentsline {subsection}{A15\ \ VertexCover ist NP-vollständig}{23}{section*.16}%
|
||||
\contentsline {subsection}{A16\ \ $k$-CliqueUniversal ist NP-vollständig}{25}{section*.17}%
|
||||
\contentsline {subsection}{A17\ \ Clique-Nomember ist NP-vollständig}{27}{section*.18}%
|
||||
\contentsline {subsection}{A18\ \ CliqueAndIndependentSet ist NP-schwer}{29}{section*.19}%
|
||||
\contentsline {subsection}{A19\ \ $k$-CLIQUE-DEG-3 ist NP-vollständig}{31}{section*.20}%
|
||||
\contentsline {subsection}{A20\ \ FeedbackVertexSet ist NP-vollständig}{34}{section*.21}%
|
||||
\contentsline {subsection}{A21\ \ $\Delta $-Cover ist NP-vollständig}{36}{section*.22}%
|
||||
\contentsline {subsection}{A22\ \ 3-COLOR mit Minimalgrad 3 ist NP-vollständig}{38}{section*.23}%
|
||||
\contentsline {subsection}{A23\ \ $k$-COLOR-PRECOLORING ist NP-vollständig}{40}{section*.24}%
|
||||
\contentsline {subsection}{A24\ \ HamiltonianPath $\leq _{p}$ HamiltonianCycle}{42}{section*.25}%
|
||||
\contentsline {subsection}{A25\ \ HamiltonianCycle $\leq _{p}$ HamiltonianPath}{44}{section*.26}%
|
||||
\contentsline {subsection}{A26\ \ Hitchhiker's-HamiltonianCycle ist NP-schwer}{46}{section*.27}%
|
||||
\contentsline {subsection}{A27\ \ SubsetSumCardinality ist NP-vollständig}{48}{section*.28}%
|
||||
\contentsline {subsection}{A28\ \ $(a_1{=}1)$-SubsetSum ist NP-schwer}{49}{section*.29}%
|
||||
\contentsline {subsection}{A29\ \ AtMostTwoPerSize-SubsetSum ist NP-schwer}{50}{section*.30}%
|
||||
\contentsline {subsection}{A30\ \ SubsetSum mit Teilbarkeit ist NP-vollständig}{52}{section*.31}%
|
||||
\contentsline {subsection}{A31\ \ SubsetSum ohne Zweierpotenzen ist NP-vollständig}{53}{section*.32}%
|
||||
\contentsline {subsection}{A32\ \ TSP ist nicht approximierbar}{54}{section*.33}%
|
||||
\contentsline {subsection}{A33\ \ HALT$_{\text {TM}}$ ist NP-schwer}{55}{section*.34}%
|
||||
\contentsline {section}{\numberline {4}ETH-Schranken}{56}{section.4}%
|
||||
\contentsline {subsection}{A34\ \ Lower Bounds: VertexCover}{56}{section*.35}%
|
||||
\contentsline {subsection}{A35\ \ Lower Bounds: HittingSet}{58}{section*.36}%
|
||||
\contentsline {subsection}{A36\ \ Lower Bounds entlang der Reduktionskette}{60}{section*.37}%
|
||||
\contentsline {subsection}{A37\ \ Lower Bounds: $k$-Color}{62}{section*.38}%
|
||||
\contentsline {subsection}{A38\ \ Lower Bounds: $2\,|\,\mathrm {prec}, p_i \in \{1,2\}\,|\,C_{\max }$}{64}{section*.39}%
|
||||
\contentsline {subsection}{A39\ \ Lower Bounds: DominatingSet}{65}{section*.40}%
|
||||
\contentsline {subsection}{A40\ \ Lower Bounds: GridTiling}{66}{section*.41}%
|
||||
\contentsline {subsection}{A41\ \ Lower Bounds: SubsetSum (strenge Reduktion)}{67}{section*.42}%
|
||||
\contentsline {subsection}{A42\ \ Lower Bounds: SetCover}{68}{section*.43}%
|
||||
\contentsline {subsection}{A43\ \ Lower Bounds: ILP-Feasibility}{69}{section*.44}%
|
||||
\contentsline {subsection}{A44\ \ Lower Bounds: NAE-4-SAT}{70}{section*.45}%
|
||||
\contentsline {subsection}{A45\ \ Lower Bounds: SetSplitting}{71}{section*.46}%
|
||||
\contentsline {subsection}{A46\ \ Lower Bounds: CoverClique}{73}{section*.47}%
|
||||
\contentsline {subsection}{A47\ \ Lower Bounds: $\Delta $Cover}{74}{section*.48}%
|
||||
\contentsline {section}{\numberline {5}Approximation: anwenden, Güte beweisen, Worst Case}{75}{section.5}%
|
||||
\contentsline {subsection}{A48\ \ Knapsack: Greedy und ModifiedGreedy}{75}{section*.49}%
|
||||
\contentsline {subsection}{A49\ \ LPT anwenden}{77}{section*.50}%
|
||||
\contentsline {subsection}{A50\ \ ListScheduling: scharfe Worst-Case-Instanzen}{78}{section*.51}%
|
||||
\contentsline {subsection}{A51\ \ 2ApproxVC hat Güte 2}{79}{section*.52}%
|
||||
\contentsline {subsection}{A52\ \ MAX-3-SAT: Güte 2 + scharfe Instanz}{80}{section*.53}%
|
||||
\contentsline {subsection}{A53\ \ ApproximateSubsetSum: Worst Case + Güte 2}{81}{section*.54}%
|
||||
\contentsline {subsection}{A54\ \ RoundRobin: Güte 2 + Worst Case}{82}{section*.55}%
|
||||
\contentsline {subsection}{A55\ \ Min-Edge-Cover: Güte 2 + Worst Case im Kreis}{83}{section*.56}%
|
||||
\contentsline {subsection}{A56\ \ Christofides-Rate $3/2$ ist scharf}{84}{section*.57}%
|
||||
\contentsline {subsection}{A57\ \ Christofides anwenden ($K_4$)}{85}{section*.58}%
|
||||
\contentsline {subsection}{A58\ \ $\Delta $TSP1 anwenden + Verständnisfragen}{86}{section*.59}%
|
||||
\contentsline {subsection}{A59\ \ Parallel-Task-Scheduling}{88}{section*.60}%
|
||||
\contentsline {subsection}{A60\ \ Strip Packing: NFDH}{90}{section*.61}%
|
||||
\contentsline {section}{\numberline {6}Wahr oder falsch?}{92}{section.6}%
|
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\contentsline {subsection}{A61\ \ Fragen über Fragen I}{92}{section*.62}%
|
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\contentsline {subsection}{A62\ \ Fragen über Fragen II}{93}{section*.63}%
|
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\contentsline {subsection}{A63\ \ Wenn die ETH fehlschlägt, gilt $P = NP$?}{94}{section*.64}%
|
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\contentsline {section}{\numberline {7}Turingmaschinen}{95}{section.7}%
|
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\contentsline {subsection}{A64\ \ TM für Palindrome}{95}{section*.65}%
|
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\contentsline {subsection}{A65\ \ TM für $L = \{0^{2^n}\}$ + RAM-Vergleich}{96}{section*.66}%
|
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43
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301
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lernmaterial/aufgabentypen.tex
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||||
\documentclass[11pt]{article}
|
||||
\input{style_guide.tex}
|
||||
|
||||
\linespread{1.05}
|
||||
\setlength{\parskip}{0.4em}
|
||||
|
||||
\title{\textbf{\Huge Die 7 Aufgabentypen}\\[0.5em]
|
||||
\large Musterlösungen, Rezepte und Bewertungshinweise\\
|
||||
\normalsize \textit{Analyse von Algorithmen und Komplexität} -- CAU Kiel, SS 2026\\[0.2em]
|
||||
\normalsize Schritt 3: die Muster kennen und richtig anwenden}
|
||||
\author{}
|
||||
\date{}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\thispagestyle{empty}
|
||||
|
||||
\begin{center}\itshape\large
|
||||
Jeder Klausur-Aufgabentyp einmal vorgemacht --\\
|
||||
mit Rezept, kompletter Musterlösung und dem, worauf der Korrektor achtet.
|
||||
\end{center}
|
||||
|
||||
\vfill
|
||||
|
||||
\noindent\textbf{Wofür dieses Heft.}
|
||||
Du hast das Fundament verstanden. Jetzt lernst du, es \emph{anzuwenden}. Alle
|
||||
bisherigen Klausuren, Haus- und Präsenzaufgaben lassen sich auf \textbf{sieben
|
||||
Aufgabentypen} zurückführen. Zu jedem zeigt dieses Heft: ein \emph{Rezept}
|
||||
(Schritt für Schritt), eine \emph{vollständig gelöste} Musteraufgabe im richtigen
|
||||
Stil, die \emph{Bewertungslogik} und die \emph{häufigen Fehler}.
|
||||
|
||||
\medskip
|
||||
\noindent\textbf{Die Reihenfolge ist Absicht.}
|
||||
Die Typen bauen aufeinander auf. Arbeite sie von vorne nach hinten durch:
|
||||
|
||||
\begin{center}\small
|
||||
\begin{tabular}{rll}
|
||||
\toprule
|
||||
\# & Aufgabentyp & baut auf \\
|
||||
\midrule
|
||||
1 & NP-Zugehörigkeit zeigen & Grundbaustein\\
|
||||
2 & Reduktion konstruieren (NP-Schwere) & braucht 1\\
|
||||
3 & Wahr oder falsch? & festigt 1+2\\
|
||||
4 & ETH-Schranke herleiten & braucht 2\\
|
||||
5 & Approximationsalgorithmus anwenden & neues Teilgebiet\\
|
||||
6 & Güte beweisen + Worst-Case-Instanz & braucht 5\\
|
||||
7 & Turingmaschine entwerfen & eigenständig\\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\medskip
|
||||
\noindent\textbf{Der Stil ist echt.}
|
||||
Die Musterlösungen sind im Stil bewerteter Abgaben gehalten: knapp, formal,
|
||||
jeder Schritt sichtbar begründet. Die Bewertungshinweise stammen aus echten
|
||||
Punkteschemata und Korrektor-Anmerkungen.
|
||||
|
||||
\vfill
|
||||
\newpage
|
||||
\tableofcontents
|
||||
\newpage
|
||||
|
||||
% ##################################################################
|
||||
\section{NP-Zugehörigkeit zeigen}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Der erste und einfachste Typ. Du zeigst, dass ein Problem in $\NP$ liegt --
|
||||
indem du ein Zertifikat angibst und beschreibst, wie man es prüft. Dieser
|
||||
Baustein steckt später in \emph{jedem} NP-Vollständigkeitsbeweis.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Woran du diesen Typ erkennst}
|
||||
|
||||
Die Aufgabe klingt so (wörtlich aus der Klausur SS23): \glqq Zeigen Sie auf
|
||||
\emph{verschiedene Weisen}, dass das Problem in NP liegt: (a) geben Sie ein
|
||||
Zertifikat an, (b) beschreiben Sie einen Verifizierer mit konkreter Laufzeit,
|
||||
(c) beschreiben Sie in Worten eine nichtdeterministische Turingmaschine.\grqq
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{Der NP-Dreischritt}
|
||||
\textbf{(a) Zertifikat.} Nenne die \glqq Lösung\grqq{} als Objekt (Teilmenge,
|
||||
Belegung, Reihenfolge). Sage, wie sie kodiert ist und dass sie \emph{polynomiell
|
||||
lang} ist.\\[0.4em]
|
||||
\textbf{(b) Verifizierer.} Ein deterministischer Algorithmus, der (Instanz,
|
||||
Zertifikat) bekommt. Hake drei Punkte ab:
|
||||
\begin{enumerate}
|
||||
\item \emph{Beschreibung}: Welche Prüfungen laufen in welcher Reihenfolge?
|
||||
\item \emph{Korrektheit -- beide Richtungen}: Ja-Instanz $\Rightarrow$ es gibt ein
|
||||
akzeptiertes Zertifikat; Nein-Instanz $\Rightarrow$ keines wird akzeptiert.
|
||||
\item \emph{Laufzeit}: konkret in $O$-Notation, nicht nur \glqq polynomiell\grqq.
|
||||
\end{enumerate}
|
||||
\textbf{(c) NTM.} \glqq Rate und prüfe.\grqq{} Rate das Zertifikat Stück für
|
||||
Stück nichtdeterministisch, prüfe dann deterministisch wie in (b).
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe}
|
||||
|
||||
\begin{aufgabe}{Longest Path (Klausur SS23, Aufgabe 2)}
|
||||
Gegeben ein Graph $G = (V,E)$ und eine Zahl $k$. Gibt es einen \emph{einfachen}
|
||||
Pfad (kein Knoten doppelt) der Länge $\ge k$? Zeigen Sie auf drei Weisen, dass
|
||||
\problem{Longest Path} in $\NP$ liegt.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{(a) Zertifikat.} Eine Folge von $k+1$ paarweise verschiedenen Knoten
|
||||
$(v_0, v_1, \dots, v_k)$ -- der behauptete Pfad. Jeder Knoten braucht
|
||||
$O(\log |V|)$ Bits, die Folge also $O(k \log |V|)$ Bits. Das ist polynomiell in
|
||||
der Eingabegröße.
|
||||
|
||||
\medskip
|
||||
\textbf{(b) Verifizierer.} Er bekommt $(G, k)$ und ein Zertifikat
|
||||
$(v_0, \dots, v_k)$ und prüft:
|
||||
\begin{itemize}
|
||||
\item ob alle $v_i$ \emph{paarweise verschieden} sind -- $O(k^2)$ Vergleiche
|
||||
(oder $O(k \log k)$ durch Sortieren);
|
||||
\item ob $\{v_i, v_{i+1}\} \in E$ für alle $i = 0, \dots, k-1$ gilt -- das sind
|
||||
$k$ Nachschläge in der Adjazenzmatrix zu je $O(1)$, also $O(k)$.
|
||||
\end{itemize}
|
||||
Akzeptiere genau dann, wenn beide Prüfungen bestehen.
|
||||
|
||||
\emph{Korrektheit (beide Richtungen).} Hat $G$ einen einfachen Pfad der Länge
|
||||
$\ge k$, so bilden seine ersten $k+1$ Knoten ein Zertifikat, das akzeptiert
|
||||
wird. Wird umgekehrt ein Zertifikat akzeptiert, so ist es eine Folge verschiedener
|
||||
Knoten mit lauter echten Kanten -- also ein einfacher Pfad der Länge $k$.
|
||||
|
||||
\emph{Laufzeit.} Gesamt $O(k^2) \subseteq O(|V|^2)$, polynomiell. Somit ist dies
|
||||
ein polynomieller Verifizierer, und $\problem{Longest Path} \in \NP$.
|
||||
|
||||
\medskip
|
||||
\textbf{(c) NTM.} Rate nacheinander $k+1$ Knoten auf ein Arbeitsband
|
||||
($O(k \log |V|)$ nichtdeterministische Schritte). Prüfe danach
|
||||
\emph{deterministisch} die Verschiedenheit und die Kantenbedingung wie in (b).
|
||||
Laufzeit: Raten $O(k \log |V|)$ plus Prüfen $\poly(|V|)$ -- insgesamt
|
||||
polynomiell.
|
||||
\hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Typisches Punkteschema pro Teil: \emph{Beschreibung} 2\,P., \emph{Korrektheit}
|
||||
1\,P., \emph{Laufzeit} 1\,P. Der Korrektor will sehen:
|
||||
\begin{itemize}
|
||||
\item eine \emph{konkrete} $O$-Notation -- \glqq polynomiell\grqq{} allein reicht
|
||||
nicht;
|
||||
\item \emph{beide} Korrektheitsrichtungen, nicht nur \glqq Pfad $\Rightarrow$
|
||||
Zertifikat\grqq;
|
||||
\item dass die \emph{Kardinalität} ($|C| \le k$ bzw.\ Länge $\ge k$) mitgeprüft
|
||||
wird.
|
||||
\end{itemize}
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
Die drei Klassiker: (1) nur \glqq polynomiell\grqq{} schreiben, wo eine konkrete
|
||||
Schranke verlangt ist. (2) Die Rückrichtung der Korrektheit weglassen. (3) Beim
|
||||
Verifizierer vergessen, die Größe zu prüfen -- ein Zertifikat könnte sonst
|
||||
\glqq zu klein\grqq{} oder \glqq zu groß\grqq{} sein und würde trotzdem akzeptiert.
|
||||
\end{stolper}
|
||||
|
||||
\subsection{Zweite Variante (kurz): Knapsack $\in$ NP}
|
||||
|
||||
Damit du das Muster auf ein Zahlenproblem überträgst:
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{Zertifikat:} eine Teilmenge $S \subseteq \{1,\dots,n\}$ der Gegenstände
|
||||
(ein Bit pro Gegenstand). \textbf{Verifizierer:} berechne $\sum_{i \in S} w_i$
|
||||
und $\sum_{i \in S} p_i$, prüfe $\sum_{i \in S} w_i \le B$ und
|
||||
$\sum_{i \in S} p_i \ge P$; akzeptiere entsprechend. Beide Summen und Vergleiche
|
||||
in $O(n)$. \textbf{Korrektheit:} Existiert eine zulässige Auswahl mit genügend
|
||||
Profit, wird sie akzeptiert; sonst keine. Also $\problem{Knapsack} \in \NP$.
|
||||
\hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
% ##################################################################
|
||||
\section{Reduktion konstruieren (NP-Schwere)}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Der zentrale und häufigste Klausurtyp. Du beweist, dass ein \emph{neues} Problem
|
||||
schwer ist, indem du ein bekannt schweres Problem darauf reduzierst. Fast immer
|
||||
ist das neue Problem eine \emph{Variante} eines Standardproblems.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Woran du diesen Typ erkennst}
|
||||
|
||||
\glqq Zeigen Sie, dass \problem{X} NP-vollständig (oder NP-schwer) ist.\grqq{}
|
||||
Meist ist \problem{X} ein leicht abgewandeltes bekanntes Problem -- mit einer
|
||||
Zusatzeigenschaft, gerichtet statt ungerichtet, mit einer Nebenbedingung.
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{NP-Vollständigkeit in 6 Schritten}
|
||||
\textbf{1. $\in$ NP.} Verifizierer angeben (oft: \glqq wie beim Originalproblem\grqq).
|
||||
Bei reiner NP-\emph{Schwere} entfällt dieser Schritt.\\[0.3em]
|
||||
\textbf{2. Quellproblem wählen.} Ein bekanntes NP-vollständiges $X$, das dem Ziel
|
||||
$Y$ ähnlich ist. \textbf{Richtung: $X \redp Y$} (bekannt $\redp$ neu).\\[0.3em]
|
||||
\textbf{3. Konstruktion.} Wie wird aus einer $X$-Instanz eine $Y$-Instanz?
|
||||
Präzise, mit \emph{allen} Parametern -- auch die Schranke $k$ anpassen!\\[0.3em]
|
||||
\textbf{4. Polynomialität.} Ein Satz: \glqq Die Konstruktion läuft in
|
||||
$O(\cdot)$.\grqq\\[0.3em]
|
||||
\textbf{5. Korrektheit, beide Richtungen.} $\Rightarrow$: Ja-Instanz von $X$
|
||||
liefert Ja-Instanz von $Y$. $\Leftarrow$: Ja-Instanz von $Y$ liefert Ja-Instanz
|
||||
von $X$.\\[0.3em]
|
||||
\textbf{6. Schlusssatz.} \glqq Da $X$ NP-vollständig, $X \redp Y$ und
|
||||
$Y \in \NP$: $Y$ ist NP-vollständig.\grqq
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe 1}
|
||||
|
||||
\begin{aufgabe}{Feedback Vertex Set (Hausaufgabe 10.1)}
|
||||
\problem{Feedback Vertex Set (FVS)}: Gegeben ein \emph{gerichteter} Graph
|
||||
$G = (V,E)$ und $k$. Gibt es $X \subseteq V$ mit $|X| \le k$, sodass
|
||||
$G \setminus X$ \emph{kreisfrei} ist? Zeigen Sie: \problem{FVS} ist
|
||||
NP-vollständig.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{1. $\in$ NP.} Zertifikat $X \subseteq V$. Der Verifizierer prüft
|
||||
$|X| \le k$ und ob $G \setminus X$ kreisfrei ist -- Letzteres mit dem
|
||||
Algorithmus zur \emph{topologischen Sortierung} (findet genau dann eine
|
||||
Sortierung, wenn der Graph kreisfrei ist). Beides in $O(|V| + |E|)$, also
|
||||
$\problem{FVS} \in \NP$.
|
||||
|
||||
\medskip
|
||||
\textbf{2. Quellproblem.} \problem{Vertex Cover} ist NP-vollständig. Wir zeigen
|
||||
$\problem{VC} \redp \problem{FVS}$.
|
||||
|
||||
\medskip
|
||||
\textbf{3. Konstruktion.} Sei $(G = (V,E), k)$ eine \problem{VC}-Instanz
|
||||
(ungerichtet). Baue den gerichteten Graphen $G' = (V, E')$ mit
|
||||
\[
|
||||
E' = \{\, (u,v),\ (v,u) \ \mid\ \{u,v\} \in E \,\},
|
||||
\]
|
||||
also: \emph{jede ungerichtete Kante wird zu zwei antiparallelen Bögen} (einem
|
||||
gerichteten Kreis der Länge 2). Die Schranke bleibt $k' = k$. Ausgabe $(G', k)$.
|
||||
|
||||
\begin{center}
|
||||
\setlength{\unitlength}{1cm}
|
||||
\begin{picture}(9,1.7)(0,-0.4)
|
||||
\put(0.6,0.5){\circle*{0.14}}\put(0.3,0.7){\small $u$}
|
||||
\put(2.6,0.5){\circle*{0.14}}\put(2.75,0.7){\small $v$}
|
||||
\put(0.6,0.5){\line(1,0){2}}
|
||||
\put(1.2,-0.05){\small ungerichtet in $G$}
|
||||
\put(3.3,0.45){\small $\leadsto$}
|
||||
\put(5.0,0.5){\circle*{0.14}}\put(4.7,0.7){\small $u$}
|
||||
\put(7.0,0.5){\circle*{0.14}}\put(7.15,0.7){\small $v$}
|
||||
\put(5.15,0.68){\vector(1,0){1.7}}
|
||||
\put(6.85,0.32){\vector(-1,0){1.7}}
|
||||
\put(5.0,-0.05){\small zwei antiparallele Bögen in $G'$}
|
||||
\end{picture}
|
||||
\end{center}
|
||||
|
||||
\textbf{4. Polynomialität.} Pro Kante werden zwei Bögen erzeugt, also
|
||||
$O(|E|)$ -- polynomiell.
|
||||
|
||||
\medskip
|
||||
\textbf{5. Korrektheit.}
|
||||
\begin{itemize}
|
||||
\item[$\Rightarrow$] Sei $C$ ein Vertex Cover mit $|C| \le k$. Dann berührt $C$
|
||||
jede Kante, also enthält $G \setminus C$ keine Kante. Damit enthält
|
||||
$G' \setminus C$ keinen Bogen und erst recht keinen Kreis -- $C$ ist ein FVS.
|
||||
\item[$\Leftarrow$] Sei $X$ ein FVS mit $|X| \le k$. Angenommen, $X$ wäre
|
||||
\emph{kein} Vertex Cover. Dann gäbe es eine Kante $\{u,v\} \in E$ mit
|
||||
$u \notin X$ und $v \notin X$. In $G'$ bleiben dann beide Bögen $(u,v)$ und
|
||||
$(v,u)$ übrig -- also der Kreis $(u, v, u)$ in $G' \setminus X$. Das ist ein
|
||||
Widerspruch dazu, dass $X$ ein FVS ist. Also ist $X$ ein Vertex Cover.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{6. Schluss.} Da \problem{VC} NP-vollständig ist, $\problem{VC} \redp
|
||||
\problem{FVS}$ gilt und $\problem{FVS} \in \NP$, ist \problem{FVS}
|
||||
NP-vollständig. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Musteraufgabe 2 (kurz): Clique $\redp$ Vertex Cover}
|
||||
|
||||
Die kürzeste Reduktion überhaupt -- reine Komplement-Umformung, kein Gadget.
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{Konstruktion.} Aus der \problem{Clique}-Instanz $(G, k)$ mit $n = |V|$
|
||||
mache die \problem{VC}-Instanz $(\bar G, n - k)$: nimm den Komplementgraphen und
|
||||
setze die Schranke auf $n - k$.
|
||||
|
||||
\textbf{Korrektheit.}
|
||||
$\Rightarrow$ Ist $C$ eine Clique der Größe $\ge k$ in $G$, so ist $C$ ein
|
||||
Independent Set in $\bar G$, und $V \setminus C$ ein Vertex Cover in $\bar G$
|
||||
der Größe $\le n - k$.
|
||||
$\Leftarrow$ Ist $D$ ein Vertex Cover der Größe $\le n-k$ in $\bar G$, so ist
|
||||
$V \setminus D$ ein Independent Set der Größe $\ge k$ in $\bar G$ -- also eine
|
||||
Clique der Größe $\ge k$ in $G$.
|
||||
|
||||
\textbf{Laufzeit.} Komplementgraph in $O(n^2)$. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Punkteschema (aus Hausaufgabe 11.1): Verifizierer $1{+}0{,}5{+}0{,}5$\,P.;
|
||||
Angabe der Reduktion 1\,P.; \emph{Korrektheit 3\,P.} -- davon \textbf{1,5 pro
|
||||
Richtung}; Laufzeit 1\,P.; Beweisführung 1\,P. Merke: Die Rückrichtung zu
|
||||
vergessen kostet also glatt 1,5 Punkte.
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
\textbf{Die vier teuren Fehler:}
|
||||
\begin{itemize}
|
||||
\item \emph{Falsche Richtung.} Immer \textbf{bekannt $\redp$ neu}
|
||||
($X_{\text{bekannt}} \redp Y_{\text{neu}}$), nie umgekehrt.
|
||||
\item \emph{Rückrichtung vergessen} -- kostet 1,5 von 5 Punkten.
|
||||
\item \emph{Zusatzeigenschaft nicht hergestellt.} Prüfe, dass jede konstruierte
|
||||
$Y$-Instanz die geforderte Sonderstruktur wirklich hat (universeller Knoten,
|
||||
Grad-Bedingung, \dots).
|
||||
\item \textbf{Nicht \emph{alle} Fälle in der Korrektheit.} Siehe der echte
|
||||
Punktabzug unten.
|
||||
\end{itemize}
|
||||
\end{stolper}
|
||||
|
||||
\begin{bewertung}
|
||||
\textbf{Echter Punktabzug (Hausaufgabe 11.1, 9/10).} Bei der Reduktion
|
||||
$\problem{VC} \redp \Delta\text{-}\problem{Cover}$ (jede Kante $e=\{u,v\}$ wird
|
||||
zum Dreieck $\{v_e, u, v\}$ aufgeblasen) stand im Beweis: \glqq \emph{jedes}
|
||||
Dreieck in $G'$ hat die Form $\{v_e, u, v\}$\grqq. Das ist falsch: Weil
|
||||
$E \subseteq E'$ gilt, \emph{erbt} $G'$ auch alle Dreiecke, die $G$ schon hatte
|
||||
(drei paarweise verbundene Original-Knoten). Diese haben \emph{nicht} die
|
||||
Gadget-Form. Der Beweis muss zusätzlich zeigen, dass auch so ein geerbtes
|
||||
Dreieck $\{a,b,c\}$ vom Cover getroffen wird -- was aus dem Vertex Cover folgt
|
||||
(die Kante $\{a,b\}$ ist gedeckt, also $a \in C$ oder $b \in C$).
|
||||
\emph{Lehre: In der Korrektheit immer \textbf{alle} Objekte abdecken, auch die
|
||||
aus dem Originalgraphen geerbten.}
|
||||
\end{bewertung}
|
||||
|
||||
% ##################################################################
|
||||
\section{Wahr oder falsch? Mit Begründung}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Dieser Typ testet, ob du die Begriffe wirklich beherrschst. Du bekommst mehrere
|
||||
Aussagen über P, NP, Reduktionen und Vollständigkeit und musst jede als wahr
|
||||
oder falsch \emph{begründen}. Er festigt genau das, was du in Kapitel 1 und 2
|
||||
gelernt hast.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{So knackst du jede Aussage}
|
||||
\textbf{1.} Führe die Aussage auf die \emph{Definitionen} zurück. Zwei
|
||||
Merksätze reichen fast immer:
|
||||
\begin{itemize}
|
||||
\item \emph{NP-schwer} = untere Schranke (\glqq mindestens so schwer wie alles in
|
||||
NP\grqq). \emph{$\in$ NP} = obere Schranke.
|
||||
\item $A \redp B$ heißt \glqq $B$ mindestens so schwer wie $A$\grqq. Die Richtung
|
||||
ist alles.
|
||||
\end{itemize}
|
||||
\textbf{2.} Ist die Aussage \emph{falsch}, gib ein \textbf{konkretes
|
||||
Gegenbeispiel} (oft $L = \emptyset$ oder ein triviales $L \in \Pclass$).\\
|
||||
\textbf{3.} Ist sie \emph{wahr}, begründe in einem Satz aus der Definition.
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe}
|
||||
|
||||
\begin{aufgabe}{Fragen über Fragen (Präsenz 11.1)}
|
||||
Entscheide jeweils wahr/falsch und begründe. Sei $L$ eine Sprache.
|
||||
\begin{enumerate}
|
||||
\item[(a)] $L \in \NP$ und $L \redp \problem{3-SAT}$ $\Rightarrow$ $L$ ist
|
||||
NP-vollständig.
|
||||
\item[(b)] $L \redp \problem{3-SAT}$ und $\problem{3-SAT} \redp L$ $\Rightarrow$
|
||||
$L$ ist NP-vollständig.
|
||||
\item[(c)] $L$ NP-vollständig $\Rightarrow$ ($L \in \Pclass \iff \Pclass = \NP$).
|
||||
\item[(d)] Es ist möglich, dass $\problem{3-SAT} \in \Pclass$ und
|
||||
$\problem{Clique} \notin \Pclass$.
|
||||
\end{enumerate}
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{(a) Falsch.} $L \redp \problem{3-SAT}$ heißt nur, dass sich $L$ \emph{auf}
|
||||
3-SAT reduziert -- das macht $L$ \emph{nicht} schwer, sondern höchstens
|
||||
\glqq so leicht wie 3-SAT\grqq. Gegenbeispiel: $L = \emptyset$. Es liegt in
|
||||
$\NP$ und ist auf 3-SAT reduzierbar, aber sicher nicht NP-vollständig. Für
|
||||
NP-Schwere bräuchte man die \emph{andere} Richtung $\problem{3-SAT} \redp L$.
|
||||
|
||||
\medskip
|
||||
\textbf{(b) Wahr.} Das ist genau die Definition. Aus $\problem{3-SAT} \redp L$
|
||||
folgt, dass $L$ NP-schwer ist (3-SAT ist NP-vollständig, Schwere vererbt sich).
|
||||
Aus $L \redp \problem{3-SAT}$ folgt $L \in \NP$ (denn 3-SAT $\in \NP$, und die
|
||||
Reduktion überträgt die Zugehörigkeit). Schwer und in NP $=$ NP-vollständig.
|
||||
|
||||
\medskip
|
||||
\textbf{(c) Wahr.} Ist $L$ NP-vollständig, so reduziert sich jedes
|
||||
$L' \in \NP$ auf $L$. Läge $L$ in $\Pclass$, wäre damit jedes $L' \in \Pclass$,
|
||||
also $\Pclass = \NP$. Umgekehrt folgt aus $\Pclass = \NP$ trivial
|
||||
$L \in \Pclass$. (Das ist der zentrale Satz über NP-vollständige Probleme.)
|
||||
|
||||
\medskip
|
||||
\textbf{(d) Falsch.} \problem{Clique} lässt sich auf \problem{3-SAT} reduzieren
|
||||
(3-SAT ist NP-vollständig, also ist jedes NP-Problem auf es reduzierbar). Läge
|
||||
\problem{3-SAT} in $\Pclass$, folgte über diese Reduktion auch
|
||||
$\problem{Clique} \in \Pclass$. Die beschriebene Situation kann es also nicht
|
||||
geben. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Zweite Variante (kurz): Präsenz 10.1}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{(i)} \glqq Löst eine NDTM ein Problem in Polyzeit, dann auch eine
|
||||
gewöhnliche DTM.\grqq{} -- \emph{Nicht beweisbar wahr}: Das würde $\Pclass = \NP$
|
||||
bedeuten, und das ist offen.\\[0.3em]
|
||||
\textbf{(ii)} \glqq $A$ NP-schwer $\Rightarrow A \in \Pclass$ ausgeschlossen /
|
||||
$A \in \NP$.\grqq{} -- \emph{Falsch}: NP-schwer sagt nichts über die
|
||||
Zugehörigkeit zu NP. Das \problem{Halteproblem} ist NP-schwer, liegt aber nicht
|
||||
in $\NP$ (es ist nicht einmal entscheidbar). \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Hier zählt \emph{ausschließlich die Begründung}, nicht das Kreuz. Ein richtiges
|
||||
\glqq falsch\grqq{} ohne Gegenbeispiel gibt kaum Punkte. Nenne bei falschen
|
||||
Aussagen immer ein \emph{konkretes} Gegenbeispiel; bei wahren die \emph{eine}
|
||||
Definition, aus der es folgt.
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
Der Dauerbrenner ist die \emph{Reduktionsrichtung}. \glqq $L \redp \problem{3-SAT}$\grqq{}
|
||||
macht $L$ nicht schwer -- es macht $L$ höchstens leicht. Verwechselst du die
|
||||
Richtung, drehst du fast jede Aussage ins Gegenteil.
|
||||
\end{stolper}
|
||||
|
||||
% ##################################################################
|
||||
\section{ETH-Schranke herleiten}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Die typische \emph{letzte} Klausuraufgabe. Aus einer \emph{vorgegebenen}
|
||||
Reduktion leitest du untere Laufzeitschranken her -- unter Annahme der
|
||||
Exponentialzeit-Hypothese (ETH). Du musst hier keine Reduktion erfinden, sondern
|
||||
sauber \emph{rechnen} und argumentieren.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Woran du diesen Typ erkennst}
|
||||
|
||||
\glqq Gegeben die Reduktion \dots. Zeigen Sie unter Annahme der ETH, dass
|
||||
\problem{X} nicht in Zeit $2^{o(p)} \cdot \poly$ lösbar ist -- für die Parameter
|
||||
$p \in \{\dots\}$.\grqq{} Meist sind mehrere Parameter gefragt (Universumsgröße,
|
||||
Mengenzahl, $k$), und bei manchen brauchst du das Sparsification-Lemma, bei
|
||||
anderen nicht.
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{ETH-Schranke in 3 Schritten}
|
||||
\textbf{1. Parameter durch $n$ und $m$ ausdrücken.} Drücke jede gefragte
|
||||
Instanzgröße (Knoten, Mengen, $k$, \dots) durch die Variablenzahl $n$ und die
|
||||
Klauselzahl $m$ der 3-SAT-Formel aus. Nutze $n \le 3m$.\\[0.3em]
|
||||
\textbf{2. Kontrapositions-Baustein hinschreiben.} Für jeden Parameter $p$:
|
||||
\begin{quote}\small\itshape
|
||||
Angenommen, es gäbe einen Algorithmus, der \problem{X} in
|
||||
$2^{o(p)} \cdot |I|^{O(1)}$ löst. Zusammen mit der Reduktion ergäbe das einen
|
||||
Algorithmus für 3-SAT in $2^{o(\,\cdot\,)}$. Nach der ETH (bzw.\ dem
|
||||
Sparsification-Lemma) ist das ausgeschlossen -- Widerspruch.
|
||||
\end{quote}
|
||||
\textbf{3. Das Sparsification-Lemma nur zitieren, wenn nötig.} Hängt $p$ an $n$,
|
||||
brauchst du es \emph{nicht} (die ETH spricht direkt über $n$). Hängt $p$ an $m$,
|
||||
\emph{musst} du es zitieren (die ETH allein verbietet $2^{o(m)}$ nicht).
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe}
|
||||
|
||||
\begin{aufgabe}{Lower Bounds für Hitting Set (Hausaufgabe 12.2)}
|
||||
\problem{Hitting Set}: Universum $U$, Mengen $F_1, \dots, F_r \subseteq U$,
|
||||
Zahl $k$. Gibt es $S \subseteq U$, $|S| \le k$, das jede Menge trifft
|
||||
($S \cap F_i \ne \emptyset$)? \emph{Gegeben} ist die Reduktion
|
||||
$\problem{3-SAT} \redp \problem{Hitting Set}$:
|
||||
\begin{itemize}
|
||||
\item $U = \{x_1, \bar x_1, \dots, x_n, \bar x_n\}$ (alle Literale),
|
||||
\item für jede Variable $i$ die Menge $F_i = \{x_i, \bar x_i\}$,
|
||||
\item für jede Klausel $j$ die Menge $F_{n+j} = \{\text{Literale von } C_j\}$,
|
||||
\item $k = n$.
|
||||
\end{itemize}
|
||||
Geben Sie (1) die Parameter $|U|, r, k$ an und leiten Sie (2) unter der ETH die
|
||||
unteren Schranken bezüglich $|U|$, $r$ und $k$ her.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{Teil 1 -- Parameter.} Es gilt
|
||||
\[
|
||||
|U| = 2n, \qquad r = n + m, \qquad k = n.
|
||||
\]
|
||||
|
||||
\textbf{Teil 2 -- untere Schranken.}
|
||||
|
||||
\emph{Bezüglich $|U|$: kein $2^{o(|U|)}$.} Angenommen, ein Algorithmus löste
|
||||
\problem{Hitting Set} in $2^{o(|U|)} \cdot \poly$. Wegen $|U| = 2n$ wäre das
|
||||
$2^{o(n)}$. Über die Reduktion ergäbe das einen $2^{o(n)}$-Algorithmus für
|
||||
3-SAT -- Widerspruch \emph{direkt} zur ETH. (Kein Sparsification-Lemma nötig, da
|
||||
$|U|$ nur an $n$ hängt.)
|
||||
|
||||
\medskip
|
||||
\emph{Bezüglich $r$: kein $2^{o(r)}$.} Hier ist $r = n + m$. Wegen $n \le 3m$
|
||||
gilt $r \le 4m = O(m)$. Ein $2^{o(r)}$-Algorithmus wäre also ein
|
||||
$2^{o(m)}$-Algorithmus und ergäbe über die Reduktion einen
|
||||
$2^{o(m)}$-Algorithmus für 3-SAT. Nach dem \textbf{Sparsification-Lemma} ist das
|
||||
unter der ETH ausgeschlossen -- Widerspruch. (Hier ist das Lemma
|
||||
\emph{zwingend}, weil der Parameter an $m$ hängt.)
|
||||
|
||||
\medskip
|
||||
\emph{Bezüglich $k$: kein $2^{o(k)}$.} Wegen $k = n$ ist das genau wie im ersten
|
||||
Fall: $2^{o(k)} = 2^{o(n)}$ ergäbe $2^{o(n)}$ für 3-SAT -- direkter Widerspruch
|
||||
zur ETH. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Wiederhole den Schluss-Baustein \emph{in jedem} Teilpunkt vollständig -- der
|
||||
Korrektor will pro Parameter das ganze Argument sehen, nicht nur beim ersten.
|
||||
Entscheidend ist, ob du bei jedem Parameter richtig erkennst, \emph{ob} das
|
||||
Sparsification-Lemma gebraucht wird.
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
Der Kern: \emph{$n$ oder $m$?} Hängt der Parameter an der Variablenzahl $n$,
|
||||
argumentierst du \emph{direkt} mit der ETH. Hängt er an der Klauselzahl $m$
|
||||
(oder an $n+m$, was $O(m)$ ist), \emph{musst} du das Sparsification-Lemma
|
||||
nennen. Es einfach immer oder nie zu zitieren, kostet Punkte.
|
||||
\end{stolper}
|
||||
|
||||
% ##################################################################
|
||||
\section{Approximationsalgorithmus anwenden}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Ein Themenwechsel: weg von den Härte-Beweisen, hin zur Approximation. Dieser Typ
|
||||
ist eine reine \emph{Rechenaufgabe} -- du lässt einen bekannten Algorithmus
|
||||
Schritt für Schritt auf eine konkrete Instanz laufen. Wenig Kreativität, viel
|
||||
Sorgfalt.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Woran du diesen Typ erkennst}
|
||||
|
||||
\glqq Wenden Sie den Algorithmus \dots auf die folgende Instanz an. Geben Sie
|
||||
alle Zwischenschritte an.\grqq{} Kandidaten: List Scheduling / LPT, $\Delta$TSP$_1$,
|
||||
Christofides, Knapsack-Greedy / Modified Greedy.
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{Was du flüssig können musst}
|
||||
\begin{itemize}
|
||||
\item \textbf{LPT / List Scheduling:} Jobs (bei LPT erst absteigend sortieren) der
|
||||
Reihe nach auf die Maschine mit \emph{kleinster} Last; Schedule als
|
||||
Gantt-Balken zeichnen; Güte nennen.
|
||||
\item \textbf{$\Delta$TSP$_1$:} minimaler Spannbaum $\to$ Kanten verdoppeln $\to$
|
||||
Eulerkreis $\to$ abkürzen.
|
||||
\item \textbf{Christofides:} MST $\to$ ungerade Knoten $\to$ min.\ Matching
|
||||
darauf $\to$ Eulerkreis $\to$ abkürzen. \emph{Alle Zwischengraphen angeben.}
|
||||
\item \textbf{Knapsack-Greedy / MGA:} nach Profitdichte $p_i/w_i$ sortieren,
|
||||
einpacken was passt; MGA zusätzlich mit dem besten Einzel-Item vergleichen.
|
||||
\end{itemize}
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe}
|
||||
|
||||
\begin{aufgabe}{Knapsack Greedy und Modified Greedy (Klausur SS23, Aufgabe 1)}
|
||||
Kapazität $B = 16$. Gegenstände als $(p_i, w_i)$:
|
||||
\[
|
||||
(1,4),\ (1,1),\ (1,3),\ (11,13),\ (3,3),\ (5,6),\ (1,5).
|
||||
\]
|
||||
Führen Sie Greedy (GA) und Modified Greedy (MGA) aus und vergleichen Sie mit dem
|
||||
Optimum.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{Schritt 1 -- Profitdichten $p_i/w_i$ berechnen und sortieren.}
|
||||
\begin{center}\small
|
||||
\begin{tabular}{lccccccc}
|
||||
\toprule
|
||||
$(p,w)$ & $(1,4)$ & $(1,1)$ & $(1,3)$ & $(11,13)$ & $(3,3)$ & $(5,6)$ & $(1,5)$\\
|
||||
Dichte & $0{,}25$ & $1$ & $0{,}33$ & $0{,}85$ & $1$ & $0{,}83$ & $0{,}2$\\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Absteigend sortiert: $(1,1),\ (3,3),\ (11,13),\ (5,6),\ (1,3),\ (1,4),\ (1,5)$.
|
||||
|
||||
\medskip
|
||||
\textbf{Schritt 2 -- Greedy einpacken} (Kapazität $B=16$, laufende Last):
|
||||
\begin{itemize}
|
||||
\item $(1,1)$: passt, Last $1$.
|
||||
\item $(3,3)$: passt, Last $4$.
|
||||
\item $(11,13)$: $4+13 = 17 > 16$ -- \emph{passt nicht}.
|
||||
\item $(5,6)$: $4+6 = 10$, passt, Last $10$.
|
||||
\item $(1,3)$: $10+3 = 13$, passt, Last $13$.
|
||||
\item $(1,4)$: $13+4 = 17 > 16$ -- passt nicht. $(1,5)$: $13+5 = 18$ -- passt nicht.
|
||||
\end{itemize}
|
||||
Greedy-Profit: $\mathrm{GA} = 1 + 3 + 5 + 1 = \mathbf{10}$.
|
||||
|
||||
\medskip
|
||||
\textbf{Schritt 3 -- Modified Greedy.} Vergleiche mit dem besten Einzel-Item.
|
||||
Das ist $(11,13)$ mit Profit $11 > 10$. Also $\mathrm{MGA} = \mathbf{11}$.
|
||||
|
||||
\medskip
|
||||
\textbf{Schritt 4 -- Optimum und Vergleich.} Optimal ist
|
||||
$\{(11,13), (3,3)\}$: Gewicht $13 + 3 = 16 \le B$, Profit $\OPT = \mathbf{14}$.
|
||||
Greedy verfehlt das schwere, profitable Item wegen seiner mittelmäßigen Dichte;
|
||||
MGA repariert das teilweise (garantierte Güte 2). \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Zweite Variante: Christofides mit allen Zwischenschritten}
|
||||
|
||||
\begin{aufgabe}{Christofides (Klausur SS23-N, Aufgabe 1)}
|
||||
Knoten $a,b,c,d$ mit $d(a,b)=1$, $d(b,c)=2$, $d(a,d)=2$, $d(a,c)=3$, $d(c,d)=3$,
|
||||
$d(b,d)=3$. Wenden Sie Christofides an.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{(1) MST} (Kruskal): $\{a,b\}\,(1)$, $\{b,c\}\,(2)$, $\{a,d\}\,(2)$;
|
||||
Gewicht $5$.\\
|
||||
\textbf{(2) Ungerade Grade im MST:} $a$ hat Grad 2, $b$ Grad 2, $c$ Grad 1,
|
||||
$d$ Grad 1 $\Rightarrow X = \{c, d\}$.\\
|
||||
\textbf{(3) Minimales perfektes Matching auf $X$:} nur die Kante $\{c,d\}$,
|
||||
Kosten $3$.\\
|
||||
\textbf{(4) Multigraph $T + K$:} Kanten $a\text{-}b, b\text{-}c, a\text{-}d,
|
||||
c\text{-}d$ -- alle Grade jetzt gerade. \textbf{Eulerkreis} ab $a$:
|
||||
$[a,b,c,d,a]$, Kosten $1+2+3+2 = 8$.\\
|
||||
\textbf{(5) Abkürzen:} kein Knoten kommt doppelt vor -- dieser Schritt ändert
|
||||
hier \emph{nichts} (trotzdem erwähnen!). Tour $[a,b,c,d,a]$, Länge $8$.\\
|
||||
Hier ist sogar $8 = \OPT$. Güte-Garantie: $1{,}5$; benötigt Symmetrie \emph{und}
|
||||
Dreiecksungleichung. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Punkte gibt es für die \emph{Zwischenschritte}, nicht nur das Endergebnis. Bei
|
||||
Christofides: alle Zwischengraphen angeben. Bei Scheduling: den Schedule als
|
||||
Gantt-Diagramm zeichnen. Und stets die \emph{Güte} des Verfahrens dazuschreiben.
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
Der Klassiker: den \emph{Abkürzen}-Schritt weglassen, wenn er nichts ändert --
|
||||
er gehört trotzdem hin. Und bei Greedy die Dichte-Sortierung nicht sauber
|
||||
hinschreiben, sodass die Einpack-Reihenfolge nicht nachvollziehbar ist.
|
||||
\end{stolper}
|
||||
|
||||
% ##################################################################
|
||||
\section{Güte beweisen und Worst-Case-Instanz bauen}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Der anspruchsvollste Approximationstyp, fast immer \emph{zweiteilig}: (a)
|
||||
beweise, dass ein Algorithmus eine bestimmte Güte einhält, und (b) konstruiere
|
||||
eine Instanz, die diese Schranke (fast) erreicht -- die also zeigt, dass die
|
||||
Schranke \emph{scharf} ist.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{Die zwei Teile}
|
||||
\textbf{Teil (a) -- Güte beweisen.} Meist eines von drei Mustern:
|
||||
\begin{itemize}
|
||||
\item \emph{OPT von unten abschätzen} über eine Struktur (z.\,B.\ ein Matching:
|
||||
jede Optimallösung braucht pro Matching-Kante einen eigenen Knoten).
|
||||
\item \emph{Widerspruch + Zählen} (z.\,B.\ MAX-3-SAT: jede Klausel wird von einer
|
||||
der beiden Belegungen erfüllt).
|
||||
\item \emph{Scheduling-Schranken}: $\OPT \ge \frac1m \sum p_i$ und
|
||||
$\OPT \ge p_{\max}$.
|
||||
\end{itemize}
|
||||
\textbf{Teil (b) -- Worst-Case-Instanz.}
|
||||
\begin{enumerate}
|
||||
\item Zwinge den Algorithmus zu einer lokal guten, global schlechten
|
||||
Entscheidung (Greedy-Falle).
|
||||
\item Parametrisiere die Instanz und berechne $A(I)$ und $\OPT(I)$ \emph{explizit}.
|
||||
\item Zeige $A(I)/\OPT(I) \to$ Schranke -- mit \emph{Grenzwert}.
|
||||
\end{enumerate}
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe}
|
||||
|
||||
\begin{aufgabe}{MAX-3-SAT, Güte 2 (Hausaufgabe 13.1 = Klausur SS23)}
|
||||
Algorithmus $A$: werte die Formel unter $\beta_0$ (alle Variablen \false) und
|
||||
$\beta_1$ (alle \true) aus, gib die bessere zurück. (a) Zeige
|
||||
$v(A(\varphi)) \ge \tfrac12\, v(\OPT(\varphi))$. (b) Gib eine Formel an, bei der
|
||||
$A$ genau die Hälfte des Optimums erreicht.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{(a) Güte-Beweis (Widerspruch + Zählen).} Sei $\varphi$ eine Formel mit
|
||||
$m$ Klauseln. \emph{Schlüsselbeobachtung:} Jede Klausel enthält mindestens ein
|
||||
Literal -- ist es positiv, wird die Klausel von $\beta_1$ (alle \true) erfüllt;
|
||||
ist es negativ, von $\beta_0$ (alle \false). Jede Klausel wird also von
|
||||
\emph{mindestens einer} der beiden Belegungen erfüllt. Summiert man:
|
||||
\[
|
||||
v(\beta_0) + v(\beta_1) \;\ge\; m.
|
||||
\]
|
||||
Also erfüllt die bessere der beiden mindestens $m/2$ Klauseln:
|
||||
$v(A(\varphi)) = \max\{v(\beta_0), v(\beta_1)\} \ge \tfrac{m}{2}$. Da eine
|
||||
optimale Belegung höchstens alle $m$ Klauseln erfüllt, ist
|
||||
$v(\OPT(\varphi)) \le m$. Zusammen:
|
||||
\[
|
||||
v(A(\varphi)) \;\ge\; \tfrac{m}{2} \;\ge\; \tfrac{1}{2}\, v(\OPT(\varphi)).
|
||||
\]
|
||||
Somit hat $A$ Güte 2.
|
||||
|
||||
\medskip
|
||||
\textbf{(b) Scharfe Instanz.} Wähle
|
||||
\[
|
||||
\varphi = (x_1 \vee x_2 \vee x_3) \;\wedge\; (\neg x_0 \vee \neg x_2 \vee \neg x_3).
|
||||
\]
|
||||
Optimum: die Belegung $x_0 = \false$, $x_1 = x_2 = x_3 = \true$ erfüllt
|
||||
\emph{beide} Klauseln, also $\OPT = 2$. Der Algorithmus aber: $\beta_1$ (alle
|
||||
\true) erfüllt nur Klausel 1 (Klausel 2 hat lauter negative Literale, alle
|
||||
falsch); $\beta_0$ (alle \false) erfüllt nur Klausel 2. Also
|
||||
$v(A(\varphi)) = \max\{1,1\} = 1 = \tfrac12 \OPT$. Die Schranke wird exakt
|
||||
erreicht. \emph{Bauprinzip: eine Klausel nur positiv, eine nur negativ.}
|
||||
\hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Zweite Variante: Approximate Subset Sum (Klausur SS24-H)}
|
||||
|
||||
\begin{aufgabe}{Greedy für Approximate Subset Sum}
|
||||
Maximiere $\sum_{j \in S} a_j \le T$. Algorithmus GA: sortiere absteigend, nimm
|
||||
der Reihe nach, \emph{stoppe} beim ersten Element, das nicht mehr passt. (a)
|
||||
Instanzen, die Güte 2 annähern; (b) zeige $\mathrm{GA}(I) \ge \OPT(I)/2$.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{(a) Worst-Case-Familie.} Für gerades $T$: $a_1 = \tfrac{T}{2}+1$,
|
||||
$a_2 = a_3 = \tfrac{T}{2}$. GA nimmt zuerst $a_1$ (das größte) und stoppt, denn
|
||||
$a_2$ passt nicht mehr ($\tfrac T2 + 1 + \tfrac T2 > T$). Also
|
||||
$\mathrm{GA} = \tfrac T2 + 1$. Optimal sind $a_2 + a_3 = T$, also $\OPT = T$.
|
||||
Rate: $\tfrac{T}{T/2+1} \to 2$ für $T \to \infty$.
|
||||
|
||||
\medskip
|
||||
\textbf{(b) Güte-Beweis.} Sei $S$ die von GA gewählte Menge und $a_\ell$ das
|
||||
erste Element, das nicht mehr passte. Dann ist
|
||||
$\mathrm{GA}(I) + a_\ell > T \ge \OPT(I)$. Wegen der absteigenden Sortierung
|
||||
enthält $S$ ein Element $a_j \ge a_\ell$, also $\mathrm{GA}(I) \ge a_\ell$.
|
||||
Einsetzen:
|
||||
\[
|
||||
2\,\mathrm{GA}(I) \;\ge\; \mathrm{GA}(I) + a_\ell \;>\; T \;\ge\; \OPT(I),
|
||||
\]
|
||||
somit $\mathrm{GA}(I) \ge \OPT(I)/2$. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Beide Teile geben Punkte -- der Güte-Beweis (a) und die Konstruktion (b). Bei
|
||||
(b) reicht keine einzelne Instanz: Gib eine \emph{parametrisierte Familie} an
|
||||
und zeige den \emph{Grenzwert} $A/\OPT \to$ Schranke. $A(I)$ und $\OPT(I)$ müssen
|
||||
\emph{explizit} berechnet sein.
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
In (a) die beiden $\OPT$-Schranken vergessen (bei Scheduling: Durchschnittslast
|
||||
\emph{und} größter Job). In (b) nur eine feste Instanz angeben statt einer
|
||||
Familie mit Grenzwert -- dann ist die Schärfe nicht gezeigt.
|
||||
\end{stolper}
|
||||
|
||||
% ##################################################################
|
||||
\section{Turingmaschine entwerfen}
|
||||
% ##################################################################
|
||||
|
||||
\begin{ziel}
|
||||
Der letzte Typ steht für sich -- er braucht keine der anderen Techniken. Du
|
||||
konstruierst eine Turingmaschine für eine formale Sprache, begründest ihre
|
||||
Korrektheit und schätzt die Laufzeit ab. Manchmal kommt ein Vergleich mit dem
|
||||
RAM-Modell dazu.
|
||||
\end{ziel}
|
||||
|
||||
\subsection{Das Rezept}
|
||||
|
||||
\begin{rezept}{TM in drei Teilen}
|
||||
\textbf{1. Die Maschine in Worten.} Beschreibe das Verhalten als nummerierte
|
||||
Schritte (\glqq lies das erste Symbol, laufe nach rechts bis \dots\grqq). Keine
|
||||
formale Zustandstabelle nötig -- eine klare Wortbeschreibung genügt.\\[0.3em]
|
||||
\textbf{2. Korrektheit.} Warum akzeptiert die TM genau die Wörter der Sprache?
|
||||
Ein bis zwei Sätze über die Schleifeninvariante (\glqq nach jedem Durchlauf sind
|
||||
die äußeren Symbole abgearbeitet\grqq).\\[0.3em]
|
||||
\textbf{3. Laufzeit.} Kosten pro Durchlauf $\times$ Anzahl Durchläufe. Denk an
|
||||
die \emph{Randfälle} (leeres Wort, ein Symbol).
|
||||
\end{rezept}
|
||||
|
||||
\subsection{Musteraufgabe}
|
||||
|
||||
\begin{aufgabe}{Palindrom-Erkenner (Hausaufgabe 10.2)}
|
||||
Konstruiere eine Turingmaschine für die Sprache
|
||||
$L = \{\, w \mid w \text{ ist ein Palindrom} \,\}$ (liest sich vorwärts wie
|
||||
rückwärts). Gib Korrektheit und Laufzeit an.
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{Die Maschine.}
|
||||
\begin{enumerate}
|
||||
\item Ist das Band leer, \emph{akzeptiere} (das leere Wort ist ein Palindrom).
|
||||
\item Steht nur noch ein Buchstabe, \emph{akzeptiere}.
|
||||
\item Lies den \emph{ersten} Buchstaben, merke ihn im Zustand, laufe zum
|
||||
\emph{letzten} Buchstaben. Sind die beiden \emph{verschieden}, \emph{verwirf}.
|
||||
Sind sie gleich, ersetze beide durch ein Leersymbol $x$ (erst den letzten, dann
|
||||
laufe zurück zum ersten und lösche ihn).
|
||||
\item Gehe zu Schritt 1.
|
||||
\end{enumerate}
|
||||
|
||||
\textbf{Korrektheit.} Die Maschine vergleicht wiederholt das äußerste Paar von
|
||||
Buchstaben und streicht es weg. Sie akzeptiert genau dann, wenn bei jedem
|
||||
Vergleich beide Enden gleich waren -- also wenn $w$ symmetrisch ist. Sie endet
|
||||
in der Mitte mit einem oder keinem Buchstaben (beides Palindrome). Bei
|
||||
Ungleichheit verwirft sie sofort.
|
||||
|
||||
\textbf{Laufzeit.} Ein Durchlauf (Schritte 1--3) kostet $O(n)$: der Kopf läuft
|
||||
einmal von links nach rechts und zurück. Pro Durchlauf werden zwei Buchstaben
|
||||
gestrichen, es gibt also höchstens $\lfloor n/2 \rfloor$ Durchläufe. Insgesamt
|
||||
$O(n) \cdot O(n) = O(n^2)$. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Zweite Variante: $L = \{0^{2^n}\}$ (Präsenz 10.3)}
|
||||
|
||||
Etwas anspruchsvoller, mit RAM-Vergleich.
|
||||
|
||||
\begin{loesung}
|
||||
\textbf{Die Maschine} (akzeptiert Nullfolgen, deren Länge eine Zweierpotenz
|
||||
ist):
|
||||
\begin{enumerate}
|
||||
\item Steht genau eine $0$ auf dem Band, \emph{akzeptiere}.
|
||||
\item Laufe über das Band und streiche \emph{jede zweite} $0$ (ersetze sie
|
||||
durch $x$).
|
||||
\item War die Anzahl der $0$en \emph{ungerade} (die letzte gelesene war eine
|
||||
$0$, die nicht gepaart wurde), \emph{verwirf}.
|
||||
\item Gehe zu Schritt 1.
|
||||
\end{enumerate}
|
||||
|
||||
\textbf{Korrektheit.} Eine Zahl ist genau dann eine Zweierpotenz, wenn
|
||||
wiederholtes Halbieren bei $1$ endet, ohne je auf eine ungerade Zahl $>1$ zu
|
||||
treffen. Genau das prüft die Maschine: jedes Streichen jeder zweiten $0$
|
||||
halbiert die Anzahl; eine ungerade Anzahl $> 1$ führt zum Verwerfen.
|
||||
|
||||
\textbf{Laufzeit.} Jeder Durchlauf kostet $O(n)$, und die Anzahl halbiert sich
|
||||
jedes Mal -- also $O(\log n)$ Durchläufe. Insgesamt $O(n \log n)$.
|
||||
|
||||
\textbf{RAM-Vergleich.} Ein RAM-Modell könnte die $0$en einfach zählen
|
||||
($O(n)$) und dann in $O(\log n)$ prüfen, ob die Zahl eine Zweierpotenz ist --
|
||||
schneller, weil es beliebige Speicherzellen direkt adressiert, während die TM
|
||||
den Kopf über das Band bewegen muss. \hfill$\square$
|
||||
\end{loesung}
|
||||
|
||||
\subsection{Bewertung und häufige Fehler}
|
||||
|
||||
\begin{bewertung}
|
||||
Punkteschema (Hausaufgabe 10.2): \emph{TM-Beschreibung} 2\,P.,
|
||||
\emph{Korrektheit} 1\,P., \emph{Laufzeit} 2\,P. Die Laufzeit ist also die Hälfte
|
||||
der Punkte -- rechne sie sauber vor (Kosten pro Durchlauf $\times$ Anzahl
|
||||
Durchläufe).
|
||||
\end{bewertung}
|
||||
|
||||
\begin{stolper}
|
||||
Die Randfälle vergessen: leeres Wort, ein einzelnes Symbol. Und die Laufzeit nur
|
||||
behaupten statt herzuleiten -- gerade hier steckt die halbe Punktzahl.
|
||||
\end{stolper}
|
||||
|
||||
\end{document}
|
||||
38
lernmaterial/aufgabentypen.toc
Normal file
38
lernmaterial/aufgabentypen.toc
Normal file
@@ -0,0 +1,38 @@
|
||||
\contentsline {section}{\numberline {1}NP-Zugehörigkeit zeigen}{3}{section.1}%
|
||||
\contentsline {subsection}{\numberline {1.1}Woran du diesen Typ erkennst}{3}{subsection.1.1}%
|
||||
\contentsline {subsection}{\numberline {1.2}Das Rezept}{3}{subsection.1.2}%
|
||||
\contentsline {subsection}{\numberline {1.3}Musteraufgabe}{3}{subsection.1.3}%
|
||||
\contentsline {subsection}{\numberline {1.4}Bewertung und häufige Fehler}{4}{subsection.1.4}%
|
||||
\contentsline {subsection}{\numberline {1.5}Zweite Variante (kurz): Knapsack $\in $ NP}{5}{subsection.1.5}%
|
||||
\contentsline {section}{\numberline {2}Reduktion konstruieren (NP-Schwere)}{5}{section.2}%
|
||||
\contentsline {subsection}{\numberline {2.1}Woran du diesen Typ erkennst}{5}{subsection.2.1}%
|
||||
\contentsline {subsection}{\numberline {2.2}Das Rezept}{5}{subsection.2.2}%
|
||||
\contentsline {subsection}{\numberline {2.3}Musteraufgabe 1}{6}{subsection.2.3}%
|
||||
\contentsline {subsection}{\numberline {2.4}Musteraufgabe 2 (kurz): Clique $\leq _{p}$ Vertex Cover}{6}{subsection.2.4}%
|
||||
\contentsline {subsection}{\numberline {2.5}Bewertung und häufige Fehler}{7}{subsection.2.5}%
|
||||
\contentsline {section}{\numberline {3}Wahr oder falsch? Mit Begründung}{8}{section.3}%
|
||||
\contentsline {subsection}{\numberline {3.1}Das Rezept}{8}{subsection.3.1}%
|
||||
\contentsline {subsection}{\numberline {3.2}Musteraufgabe}{8}{subsection.3.2}%
|
||||
\contentsline {subsection}{\numberline {3.3}Zweite Variante (kurz): Präsenz 10.1}{9}{subsection.3.3}%
|
||||
\contentsline {subsection}{\numberline {3.4}Bewertung und häufige Fehler}{9}{subsection.3.4}%
|
||||
\contentsline {section}{\numberline {4}ETH-Schranke herleiten}{10}{section.4}%
|
||||
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\documentclass[10pt,a4paper]{article}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[ngerman,provide=*]{babel}
|
||||
\usepackage{amsmath,amssymb}
|
||||
\usepackage[margin=1.6cm]{geometry}
|
||||
\usepackage{xcolor}
|
||||
|
||||
\setlength{\parindent}{0pt}
|
||||
\setlength{\parskip}{0.25em}
|
||||
|
||||
\newcommand{\prob}[1]{\textsc{#1}}
|
||||
\newcommand{\redp}{\le_p}
|
||||
|
||||
% Problemblock: Name
|
||||
\newcommand{\pblock}[1]{\par\medskip\noindent\colorbox{black!8}{\parbox{\dimexpr\linewidth-2\fboxsep}{\textbf{\large #1}}}\par\nopagebreak\smallskip}
|
||||
\newcommand{\feld}[1]{\par\nopagebreak\textbf{#1:}\ }
|
||||
\newcommand{\hin}{\feld{$\Rightarrow$}}
|
||||
\newcommand{\rueck}{\feld{$\Leftarrow$}}
|
||||
\newcommand{\skizze}{\textcolor{red!70!black}{\textbf{[Skizze -- Detail selbst prüfen]}}\ }
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{center}
|
||||
{\LARGE\bfseries AAK Cheatsheet -- Problem-Arsenal}\\[2pt]
|
||||
{\small Referenz: Definition · Beispiel · Reduktion · Konstruktion · ETH · Beweisidee je Richtung}
|
||||
\end{center}
|
||||
|
||||
\textbf{Reduktionsbaum (Kurs):}
|
||||
\[
|
||||
\begin{array}{l}
|
||||
\prob{Sat} \redp \text{3-\prob{Sat}} \redp k\text{-}\prob{Clique} \redp \prob{IndependentSet},\ \prob{VertexCover} \redp \prob{FVS},\ \Delta\prob{Cover}\\[2pt]
|
||||
\text{3-\prob{Sat}} \redp \prob{HittingSet} \redp \prob{SetCover}; \quad \text{3-\prob{Sat}} \redp \prob{DominatingSet},\ k\text{-}\prob{Color} \redp \prob{CoverClique}\\[2pt]
|
||||
\prob{Sat} \redp \text{3-DM} \redp \text{3-\prob{ExactCover}} \redp \prob{SubsetSum} \redp \prob{Partition} \redp P2\,\|\,C_{\max}; \quad \prob{SubsetSum} \redp \prob{Knapsack}\\[2pt]
|
||||
\prob{VertexCover} \redp \prob{HamiltonianCycle} \leftrightarrow \prob{HamiltonianPath}; \quad \prob{HC} \redp \prob{TSP}\\[2pt]
|
||||
\text{3-\prob{Sat}} \redp \text{NAE-4-\prob{Sat}} \redp \prob{SetSplitting}; \quad k\text{-}\prob{Clique} \redp \prob{GridTiling},\ 2|\mathrm{prec}|C_{\max}
|
||||
\end{array}
|
||||
\]
|
||||
|
||||
\textbf{Quellen-Wahl bei fremdem Problem:} Zahlen/Summen $\to$ \prob{SubsetSum}/\prob{Partition} · Knoten-Auswahl $\to$ \prob{Clique}/\prob{VC}/\prob{IS} · Touren/Reihenfolgen $\to$ \prob{HC}/\prob{HP} · Färbung/Partition in Klassen $\to$ $k$-\prob{Color} · Mengensysteme $\to$ \prob{HittingSet}/\prob{3-ExactCover} · Scheduling $\to$ \prob{Partition}/$P2\|C_{\max}$ · Zusatzeigenschaft auf bekanntem Problem $\to$ vom Grundproblem reduzieren, Eigenschaft per Gadget erzwingen.
|
||||
|
||||
\hrulefill
|
||||
|
||||
\pblock{SAT}
|
||||
\feld{Def} KNF-Formel $\varphi$; existiert erfüllende Belegung?
|
||||
\feld{Bsp} $(x \vee y) \wedge (\bar x \vee y \vee z) \wedge (\bar y)$ -- ja: $x{=}0, y{=}0, z{=}1$.
|
||||
\feld{Reduktion} Basis (Cook--Levin: jede NP-Sprache $\redp$ \prob{Sat}).
|
||||
\feld{ETH} über 3-\prob{Sat}-Schranken.
|
||||
\feld{Rolle} Startpunkt; Klauseln beliebig lang -- Vorsicht: SS24-Klausur verlangte $\prob{Sat} \redp k$-\prob{Clique}, nicht 3-\prob{Sat}!
|
||||
|
||||
\pblock{3-SAT}
|
||||
\feld{Def} KNF, jede Klausel $\le 3$ Literale; erfüllbar?
|
||||
\feld{Bsp} $(x_1 \vee x_2 \vee x_3) \wedge (\bar x_1 \vee \bar x_2 \vee \bar x_3)$ -- ja: $x_1{=}1, x_2{=}0$.
|
||||
\feld{Reduktion} von \prob{Sat}.
|
||||
\feld{Konstruktion} Lange Klausel $(\ell_1 \vee \dots \vee \ell_k)$ mit frischen $y_i$ zerhacken: $(\ell_1 \vee \ell_2 \vee y_1)(\bar y_1 \vee \ell_3 \vee y_2)\dots$
|
||||
\hin Erfülltes $\ell_j$ wählen, $y$-Kette davor wahr, danach falsch setzen.
|
||||
\rueck Wären alle $\ell_j$ falsch, erzwingt die $y$-Kette einen Widerspruch (Dominoeffekt).
|
||||
\feld{ETH} nicht in $2^{o(n)}$; per \textbf{Sparsification-Lemma} auch nicht in $2^{o(m)}$. Merke: $n \le 3m$.
|
||||
|
||||
\pblock{$k$-CLIQUE}
|
||||
\feld{Def} Graph $G$, Zahl $k$; existiert paarweise verbundene Knotenmenge $|C| \ge k$?
|
||||
\feld{Bsp} Dreieck $\{1,2,3\}$ plus Kante $\{3,4\}$: $k{=}3$ ja, $k{=}4$ nein.
|
||||
\feld{Reduktion} von \prob{Sat} (Vorlesung).
|
||||
\feld{Konstruktion} Knoten = Paare (Klausel $C_j$, Literal $\ell \in C_j$); Kante zwischen Paaren verschiedener Klauseln, außer $\ell = \bar\ell'$; $k = m$.
|
||||
\hin Erfüllende Belegung: pro Klausel ein wahres Literal wählen -- paarweise kompatibel $\to$ $m$-Clique.
|
||||
\rueck $m$-Clique hat je Klausel genau einen Knoten; Literale widerspruchsfrei $\to$ Belegung erfüllt alle Klauseln.
|
||||
\feld{ETH} $|V| = O(m)$, $|E| = O(m^2)$, $k = m$ $\Rightarrow$ nicht in $2^{o(|V|)}$, $2^{o(\sqrt{|E|})}$, $2^{o(k)}$.
|
||||
|
||||
\pblock{$k$-INDEPENDENTSET}
|
||||
\feld{Def} Existieren $k$ paarweise \emph{nicht} verbundene Knoten?
|
||||
\feld{Bsp} Pfad $1{-}2{-}3{-}4$: $\{1,3\}$ oder $\{1,4\}$, $k{=}2$ ja.
|
||||
\feld{Reduktion} von $k$-\prob{Clique}.
|
||||
\feld{Konstruktion} Komplementgraph $\bar G$, gleiches $k$.
|
||||
\hin Clique in $G$ = keine Kanten in $\bar G$ = IS.
|
||||
\rueck IS in $\bar G$ = alle Paare Kanten in $G$ = Clique.
|
||||
\feld{ETH} $|V|$ gleich, $|E| \le |V|^2$ $\Rightarrow$ nicht in $2^{o(|V|)}$, $2^{o(\sqrt{|E|})}$.
|
||||
|
||||
\pblock{VERTEXCOVER}
|
||||
\feld{Def} Existiert $S$, $|S| \le k$, die jede Kante mit einem Endpunkt trifft?
|
||||
\feld{Bsp} Stern (Zentrum $z$, 4 Blätter): $S = \{z\}$, $k{=}1$ ja.
|
||||
\feld{Reduktion} von $k$-\prob{Clique} (Präsenz 10.2).
|
||||
\feld{Konstruktion} $\bar G$ und $k' = n - k$.
|
||||
\hin Clique $C$ in $G$ $\to$ $V \setminus C$ deckt $\bar G$: in $\bar G$ gibt es keine Kante innerhalb $C$.
|
||||
\rueck VC $V'$ in $\bar G$ $\to$ $V \setminus V'$ ist IS in $\bar G$ = Clique in $G$, Größe $\ge n - (n-k) = k$.
|
||||
\feld{ETH} $|V'| = |V|$, $|E'| \le |V|^2$, $k' \le |V|$ $\Rightarrow$ nicht in $2^{o(|V|)}$, $2^{o(\sqrt{|E|})}$, $2^{o(k)}$.
|
||||
|
||||
\pblock{FEEDBACKVERTEXSET (gerichtet)}
|
||||
\feld{Def} $X \subseteq V$, $|X| \le k$, sodass $G \setminus X$ kreisfrei.
|
||||
\feld{Bsp} Kreis $1 \to 2 \to 3 \to 1$: $X = \{1\}$, $k{=}1$ ja.
|
||||
\feld{Reduktion} von \prob{VertexCover} (HA 10.1).
|
||||
\feld{Konstruktion} Jede Kante $\{u,v\}$ durch beide Richtungen $(u,v), (v,u)$ ersetzen; $k$ übernehmen.
|
||||
\hin VC entfernt $\to$ keine Kanten mehr $\to$ kreisfrei.
|
||||
\rueck Wäre Kante $\{u,v\}$ ungedeckt, bildet $(u,v,u)$ einen 2-Kreis -- Widerspruch.
|
||||
|
||||
\pblock{$\Delta$COVER (Dreiecksüberdeckung)}
|
||||
\feld{Def} $C_\Delta$, $|C_\Delta| \le k$: jedes Dreieck enthält einen Knoten aus $C_\Delta$.
|
||||
\feld{Bsp} Zwei Dreiecke mit gemeinsamer Kante $\{u,v\}$: $C_\Delta = \{u\}$, $k{=}1$ ja.
|
||||
\feld{Reduktion} von \prob{VertexCover} (HA 11.1).
|
||||
\feld{Konstruktion} Pro Kante $e = \{u,v\}$ neuer Knoten $v_e$ mit Kanten zu $u, v$ -- jede Kante wird Dreieck; $k$ übernehmen.
|
||||
\hin VC trifft jede Kante, also jedes Alt-Dreieck \emph{und} jedes neue Dreieck $\{u, v, v_e\}$. \textbf{Achtung Korrektur H11:} auch Dreiecke, die schon in $G$ lagen, explizit abdecken!
|
||||
\rueck ObdA $C_\Delta \subseteq V$ ($v_e$ durch $e_1$ tauschen, deckt weiter); dann trifft $C_\Delta$ jedes $\{u,v,v_e\}$, also jede Kante.
|
||||
\feld{ETH} $|V'| = |V| + |E| \le 2|V|^2$, $|E'| = 3|E|$ $\Rightarrow$ nicht in $2^{o(\sqrt{|V'|})}$, $2^{o(\sqrt{|E'|})}$ (aus VC-Schranken).
|
||||
|
||||
\pblock{HAMILTONIANCYCLE (HC)}
|
||||
\feld{Def} Kreis, der jeden Knoten genau einmal besucht?
|
||||
\feld{Bsp} $C_5$ (Fünfeck): ja. $K_{1,3}$ (Stern): nein.
|
||||
\feld{Reduktion} von \prob{VertexCover} (Vorlesung) -- \textbf{Konstruktion nie selbst gefordert. Nur als Fakt zitieren.}
|
||||
\feld{Als Quelle} für \prob{HP} $\leftrightarrow$ \prob{HC}, \prob{TSP}, Hitchhiker's, \prob{LongestPath} ($k = n{-}1$).
|
||||
|
||||
\pblock{HAMILTONIANPATH (HP) $\leftrightarrow$ HC}
|
||||
\feld{Def} Pfad, der jeden Knoten genau einmal besucht?
|
||||
\feld{HP von HC} Wähle $v$; füge Kopie $v^*$ (gleiche Nachbarn) plus Grad-1-Knoten $s$ an $v$, $t$ an $v^*$ an.
|
||||
\hin HC bei $v$ aufschneiden $\to$ Pfad $s, v, \dots, v^*, t$.
|
||||
\rueck Pfad muss $s, t$ als Enden haben; $v \dots v^*$ zusammenkleben ergibt HC.
|
||||
\feld{HC von HP} Neuen Universalknoten $u$ (mit allen verbunden) hinzufügen.
|
||||
\hin Pfad $+$ $u$ am Anfang/Ende schließt Kreis.
|
||||
\rueck HC ohne $u$ ist Pfad über alle Altknoten.
|
||||
|
||||
\pblock{TSP (Entscheidung) + Inapproximierbarkeit}
|
||||
\feld{Def} Vollständiger Graph mit Distanzen, Budget $T$; Rundreise $\le T$?
|
||||
\feld{Bsp} 4 Städte im Quadrat, Seitenlänge 1: OPT $= 4$.
|
||||
\feld{Reduktion} von \prob{HC}.
|
||||
\feld{Konstruktion} $d(u,v) = 1$ falls $\{u,v\} \in E$, sonst $2$; $T = n$. (Erfüllt sogar $\Delta$-Ungleichung!)
|
||||
\hin HC $\to$ Tour aus lauter 1er-Kanten, Länge $n$.
|
||||
\rueck Tour $\le n$ kann keine 2er-Kante nutzen $\to$ alle Kanten in $E$ $\to$ HC.
|
||||
\feld{Inapprox (SS23!)} Nicht-Kanten stattdessen $= \alpha n + 1$. Ein $\alpha$-Approx-Algorithmus gäbe Tour $\le \alpha n$ gdw.\ HC existiert $\to$ entscheidet \prob{HC} in Polyzeit $\to$ $P = NP$. Für kein $\alpha$ Güte möglich, außer $P = NP$.
|
||||
|
||||
\pblock{$k$-COLOR}
|
||||
\feld{Def} Färbung $f: V \to [k]$ mit $f(u) \ne f(v)$ für Kanten?
|
||||
\feld{Bsp} $C_5$: 3 Farben nötig; $C_4$: 2 reichen.
|
||||
\feld{Reduktion} von 3-\prob{Sat} -- \textbf{in Klausuren stets gegeben, nie selbst konstruieren.} Nur Parameter zählen!
|
||||
\feld{ETH} nicht in $2^{o(|V|)}$ (stand SS25 als Hinweis dabei); Parameter der SS24-Reduktion: $|V| = O(n + m)$, $|E| = O(n^2 + nm)$ $\Rightarrow$ nicht in $2^{o(\sqrt{|E|})}$.
|
||||
\feld{Als Quelle} NP-vollständig zitieren für: \prob{Precoloring}, \prob{CoverClique}, 3-\prob{Color}-Varianten.
|
||||
|
||||
\pblock{$k$-COLOR-PRECOLORING}
|
||||
\feld{Def} Zusätzlich: vorgegebene Knoten $v_1, \dots, v_k$ mit $f(v_i) = i$.
|
||||
\feld{Reduktion} von $k$-\prob{Color}.
|
||||
\feld{Konstruktion} Füge frische Clique $v_1, \dots, v_k$ hinzu (untereinander verbunden, sonst isoliert); precolore $f(v_i) = i$.
|
||||
\hin $k$-Färbung von $G$ bleibt gültig; neue Clique bekommt Farben $1..k$ per Definition.
|
||||
\rueck Einschränkung der Precoloring-Färbung auf $G$ ist $k$-Färbung.
|
||||
\feld{Merke} Verifizierer braucht $v_1, \dots, v_k$ als Teil der \emph{Eingabe} (Klausur-Korrektur!) und muss ALLE Knoten prüfen, nicht nur $C$.
|
||||
|
||||
\pblock{COVERCLIQUE}
|
||||
\feld{Def} $k$ paarweise disjunkte Cliquen, die $V$ überdecken?
|
||||
\feld{Bsp} $C_4$ (Viereck): 2 Cliquen (gegenüberliegende Kanten) -- ja für $k{=}2$.
|
||||
\feld{Reduktion} von $k$-\prob{Color} (SS25-Klausur).
|
||||
\feld{Konstruktion} Komplementgraph $\bar G$, gleiches $k$.
|
||||
\hin Farbklassen sind IS in $G$ = Cliquen in $\bar G$; disjunkt, überdecken $V$.
|
||||
\rueck Cliquen in $\bar G$ = IS in $G$ $\to$ als Farbklassen nutzen.
|
||||
\feld{ETH} $|V|$ gleich $\Rightarrow$ nicht in $2^{o(|V|)}$; $|\bar E| \le |V|^2$ $\Rightarrow$ nicht in $2^{o(\sqrt{|E|})}$.
|
||||
|
||||
\pblock{3-DIMENSIONALMATCHING (3-DM)}
|
||||
\feld{Def} Disjunkte Mengen $W, X, Y$ ($|W|{=}|X|{=}|Y|{=}q$), Tripel $T \subseteq W {\times} X {\times} Y$; $q$ disjunkte Tripel, die alles überdecken?
|
||||
\feld{Bsp} Ehe-Analogie mit 3 Geschlechtern: perfekte Dreier-Zuordnung.
|
||||
\feld{Reduktion} von \prob{Sat} (Vorlesung; auf 3-\prob{Sat}-Instanzen anwenden für ETH) -- \textbf{Konstruktion nie selbst gefordert. Nur Parameter und Kette merken.}
|
||||
\feld{ETH} $|W| = O(mn) \le O(m^2)$, $|T| = O(m^2 n^2) \le O(m^4)$ $\Rightarrow$ nicht in $2^{o(\sqrt{|W|})}$, $2^{o(\sqrt[4]{|T|})}$.
|
||||
\feld{Rolle} Glied der Kette zu \prob{X3C} $\to$ \prob{SubsetSum} (Präsenz 13.1).
|
||||
|
||||
\pblock{3-EXACTCOVER (X3C)}
|
||||
\feld{Def} Grundmenge $U$ ($|U| = 3q$), 3er-Mengen $F$; exakte Überdeckung durch disjunkte Mengen?
|
||||
\feld{Reduktion} von 3-DM (Spezialfall: Tripel als 3er-Mengen über $U = W \dot\cup X \dot\cup Y$).
|
||||
\hin/\rueck identisch -- 3-DM \emph{ist} eine X3C-Instanz.
|
||||
\feld{ETH} $|U| = 3q$, $|F| = |T|$ $\Rightarrow$ nicht in $2^{o(\sqrt{|U|})}$, $2^{o(\sqrt[4]{|F|})}$.
|
||||
|
||||
\pblock{SUBSETSUM}
|
||||
\feld{Def} Größen $a_1, \dots, a_n$, Ziel $T$; Teilmenge mit Summe genau $T$?
|
||||
\feld{Bsp} $\{3, 5, 7, 11\}$, $T = 12$: ja ($5 + 7$).
|
||||
\feld{Reduktion 1} von X3C (Vorlesung): Menge $F_i \to$ Zahl $\sum_{u \in F_i} (|F|{+}1)^{u}$, $T = \sum_u (|F|{+}1)^u$ -- Basis verhindert Überträge.
|
||||
\feld{Reduktion 2} von 3-\prob{Sat} (SS23-Klausur, Basis 10): $s(a_i) = 10^{i-1} + \sum_{x_i \in C_j} 10^{n+j-1}$, analog $b_i$ für $\bar x_i$; Slack $c_j, d_j = 10^{n+j-1}$; Ziel: Variablenziffern 1, Klauselziffern 3.
|
||||
\hin Belegung: pro Variable $a_i$ oder $b_i$ wählen; Klauselziffer wird $1..3$, Slacks füllen auf 3.
|
||||
\rueck Keine Überträge (Basis groß genug) $\to$ Variablenziffer erzwingt genau ein Item pro Paar; Klauselziffer 3 braucht $\ge 1$ echtes Literal-Item (Slacks geben nur 2).
|
||||
\feld{ETH} bzgl.\ Itemzahl: nicht in $2^{o(\sqrt[4]{n})}$ (via X3C). Bzgl.\ $d$ (verschiedene Größen) und $L = \log \Delta$ (SS23!): beide $O(n + m)$ $\Rightarrow$ nicht in $2^{o(d)}$, nicht in $2^{o(L)}$.
|
||||
|
||||
\pblock{SUBSETSUM-Varianten (Padding-Trickkasten)}
|
||||
\feld{Cardinality ($|S| = n/2$)} Präsenz 11.3: $c_i' = c_i + 1$, plus $n$ Einser-Items, $T' = T + n$.
|
||||
\hin Lösung $S$ mit Einsern auf Größe $n$ auffüllen. \rueck Einser-Items gleichen Kardinalität aus, Kern bleibt.
|
||||
\feld{$(a_1 = 1)$} Alle Größen $\times 2$, ein 1-Item dazu, $T' = 2T$: Parität verhindert Nutzung der 1.
|
||||
\feld{Teilbar durch 3 oder 7} Alle Größen und $T$ mit $21$ multiplizieren.
|
||||
\feld{Ohne Zweierpotenzen} Alle $\times 3$: $3a$ ist nie $2^k$ (ungerader Faktor); $T' = 3T$.
|
||||
\feld{AtMostTwoPerSize} \textbf{Kam SS24 als eigene Aufgabe -- Konstruktion können!} \skizze Idee: Vielfachheit $c$ einer Größe binär bündeln: Items $a, 2a, 4a, \dots$ je einmal -- alle Teilsummen $0..ca$ erreichbar. Gegen Musterlösung prüfen.
|
||||
\feld{Muster} Immer: Ja-Instanzen exakt erhalten, Eigenschaft durch Skalierung/Padding erzwingen, Zieltransformation angeben.
|
||||
|
||||
\pblock{PARTITION}
|
||||
\feld{Def} Zahlen in zwei Hälften gleicher Summe teilen?
|
||||
\feld{Bsp} $\{1, 2, 3, 4\}$: $\{1,4\} | \{2,3\}$ -- ja.
|
||||
\feld{Reduktion} von \prob{SubsetSum} ($A = \sum a_i$).
|
||||
\feld{Konstruktion} Zwei Zusatz-Items $2A - T$ und $A + T$; Gesamtsumme $4A$, Hälfte $= 2A$.
|
||||
\hin Lösung $S$: Seite $S \cup \{2A - T\}$ hat $T + 2A - T = 2A$.
|
||||
\rueck Zusatz-Items nie zusammen ($3A > 2A$); Seite mit $2A - T$ enthält Items der Summe $T$.
|
||||
|
||||
\pblock{$P2\,\|\,C_{\max}$ (2 Maschinen)}
|
||||
\feld{Def} Jobs auf 2 Maschinen, Makespan $\le T$?
|
||||
\feld{Reduktion} von \prob{Partition}: Jobs = Zahlen, $T = \frac12 \sum p_j$.
|
||||
\hin Partition = perfekt balancierter Schedule. \rueck Makespan $\frac12\sum$ erzwingt exakte Balance.
|
||||
|
||||
\pblock{KNAPSACK (Entscheidung)}
|
||||
\feld{Def} Gewichte $w_i$, Profite $p_i$, Kapazität $K$, Zielprofit $P$.
|
||||
\feld{Reduktion} von \prob{SubsetSum}: $w_i = p_i = a_i$, $K = P = T$.
|
||||
\hin/\rueck Summe $= T$ gdw.\ Gewicht $\le T$ und Profit $\ge T$ gleichzeitig.
|
||||
|
||||
\pblock{HITTINGSET}
|
||||
\feld{Def} Universum $U$, Mengen $F_1, \dots, F_r$, Zahl $k$; $H$, $|H| \le k$, trifft jede $F_i$?
|
||||
\feld{Bsp} $U = \{1..5\}$, $F = \{1,2\}, \{2,3\}, \{4,5\}$: $H = \{2, 4\}$, $k{=}2$ ja.
|
||||
\feld{Reduktion} von 3-\prob{Sat} (HA 12.2 / Probeklausur).
|
||||
\feld{Konstruktion} $U$ = Literale; Paar-Mengen $\{x_i, \bar x_i\}$, Klausel-Mengen $C_j$; $k = n$.
|
||||
\hin Belegung $\to$ $H$ = wahre Literale: trifft jedes Paar (eines wahr) und jede Klausel (erfülltes Literal).
|
||||
\rueck $k = n$ und $n$ disjunkte Paare $\to$ genau ein Literal pro Paar = Belegung; Klauselmengen getroffen = erfüllt.
|
||||
\feld{ETH} $|U| = 2n$, $r = n + m \le 4m$, $k = n$ $\Rightarrow$ nicht in $2^{o(|U|)}$, $2^{o(r)}$ (Sparsification), $2^{o(k)}$.
|
||||
|
||||
\pblock{SETCOVER}
|
||||
\feld{Def} Mengen $F_1, \dots, F_r$ über $U$; $k$ Mengen, deren Vereinigung $U$ ist?
|
||||
\feld{Reduktion} von \prob{HittingSet} (SS23-Klausur; Rollen tauschen).
|
||||
\feld{Konstruktion} Bipartite Inzidenz: neues Universum $U' = [r]$ (alte Mengen-Indizes); pro altes Element $w$ die Menge $F_w' = \{i \mid w \in F_i\}$; $k' = k$.
|
||||
\hin Hitting-Set $H$ trifft jede $F_i$ $\leftrightarrow$ die Mengen $\{F_w' \mid w \in H\}$ überdecken $[r]$.
|
||||
\rueck symmetrisch (Dualität: Element trifft Menge $=$ Menge enthält Element).
|
||||
\feld{ETH} Parameter tauschen: $|U'| = r$, $r' = |U|$ $\Rightarrow$ \prob{SetCover} nicht in $2^{o(r)}$ und nicht in $2^{o(|U|)}$ (jeweils übersetzt).
|
||||
|
||||
\pblock{DOMINATINGSET}
|
||||
\feld{Def} $M$, $|M| = k$: jeder Knoten in $M$ oder Nachbar von $M$.
|
||||
\feld{Bsp} Stern: Zentrum dominiert alles, $k{=}1$.
|
||||
\feld{Reduktion} von 3-\prob{Sat} (SS24-Klausur).
|
||||
\feld{Konstruktion} Pro Variable Dreieck $\{x_i, \bar x_i, d_i\}$; pro Klausel Knoten $C_j$ mit Kanten zu seinen Literalen; $k = n$.
|
||||
\hin Belegung $\to$ $M$ = wahre Literale: dominiert Dreieck (alle drei benachbart) und $C_j$ (erfülltes Literal).
|
||||
\rueck $k = n$ Dreiecke brauchen je einen Knoten ($d_i$ nur intern nützlich $\to$ obdA Literal); $C_j$ dominiert $\to$ ein Literal wahr.
|
||||
\feld{ETH} $|V| = 3n + m = O(m)$, $|E| = 3n + 3m = O(m)$ (\textbf{linear!}) $\Rightarrow$ nicht in $2^{o(|V|)}$ und nicht in $2^{o(|E|)}$ -- Kanten OHNE Wurzel.
|
||||
|
||||
\pblock{NAE-4-SAT}
|
||||
\feld{Def} Klauseln $\le 4$ Literale; Belegung, sodass jede Klausel wahres UND falsches Literal hat (``not all equal'')?
|
||||
\feld{Reduktion} von 3-\prob{Sat} (SS25-Klausur).
|
||||
\feld{Konstruktion} Frische Variable $w$; jede Klausel $C_i \to C_i \cup \{w\}$.
|
||||
\hin Erfüllende Belegung $+$ $w = $ falsch: wahres Literal existiert, $w$ ist das falsche.
|
||||
\rueck NAE-Lösung; obdA $w = $ falsch (sonst alles flippen -- NAE bleibt!); dann hat jede Klausel ein wahres Original-Literal.
|
||||
\feld{ETH} $n' = n + 1$, $m' = m$ $\Rightarrow$ nicht in $2^{o(n)}$, $2^{o(m)}$.
|
||||
|
||||
\pblock{SETSPLITTING}
|
||||
\feld{Def} Grundmenge $S$, Mengen $F_i$; Partition $S = S_1 \dot\cup S_2$, sodass jede $F_i$ beide Seiten trifft?
|
||||
\feld{Reduktion} von NAE-4-\prob{Sat} (SS25-Klausur).
|
||||
\feld{Konstruktion} $S$ = Literale; Paar-Mengen $\{x_i, \bar x_i\}$ + Klausel-Mengen $C_i$.
|
||||
\hin NAE-Belegung: $S_1$ = wahre Literale. Paare gesplittet (eins wahr, eins falsch), Klauseln gesplittet (NAE!).
|
||||
\rueck Splitting: Paar-Mengen erzwingen $x_i$, $\bar x_i$ auf verschiedenen Seiten $\to$ konsistente Belegung ($S_1$ = wahr); Klausel beidseitig = NAE.
|
||||
\feld{ETH} $|S| = 2n$, $r = n + m$ $\Rightarrow$ nicht in $2^{o(|S|)}$, $2^{o(r)}$.
|
||||
|
||||
\pblock{GRIDTILING}
|
||||
\feld{Def} $k \times k$ Gitter, Mengen $S_{i,j} \subseteq [n]^2$; Auswahl $e_{i,j} \in S_{i,j}$, zeilenweise gleiche 1.\ Komponente, spaltenweise gleiche 2.\ Komponente.
|
||||
\feld{Reduktion} von \prob{Clique} (SS24-Klausur; $k \in \Theta(n)$ erlaubt).
|
||||
\feld{Konstruktion} Diagonale $S_{i,i} = \{(a,a)\}$; sonst $S_{i,j} = \{(a,b) \mid \{a,b\} \in E, a \ne b\}$; $k$ übernehmen.
|
||||
\hin Clique $v_1, \dots, v_k$: setze $e_{i,j} = (v_i, v_j)$ -- Konsistenz nach Konstruktion.
|
||||
\rueck Zeilen-/Spaltenkonsistenz definiert $v_i$ (Diagonale erzwingt Paare $(a,a)$); Off-Diagonale erzwingt Kanten zwischen allen $v_i, v_j$.
|
||||
\feld{ETH} $X = \sum |S_{i,j}| = O(k^2 |E|) = O(n^4)$, $Y = \max |S_{i,j}| = O(n^2)$ $\Rightarrow$ nicht in $2^{o(\sqrt[4]{X})}$, $2^{o(\sqrt{Y})}$.
|
||||
|
||||
\pblock{$2\,|\,\mathrm{prec},\ p \in \{1,2\}\,|\,C_{\max}$}
|
||||
\feld{Def} 2 Maschinen, Präzedenz-DAG, Zeiten 1/2; Makespan $\le T$?
|
||||
\feld{Reduktion} von $k$-\prob{Clique} (SS24-Klausur, Konstruktion gegeben).
|
||||
\feld{Konstruktion} \skizze Knoten-Jobs ($p{=}1$), Kanten-Jobs ($p{=}2$), $3|V| + 2|E|$ Dummies; Präzedenz Knoten $\to$ inzidente Kanten; $T = 2|V| + 2|E|$, $n = 4|V| + 3|E|$.
|
||||
\hin Clique-Knoten früh einplanen, ihre Kanten füllen Lücken deadline-genau.
|
||||
\rueck Enger Makespan erzwingt Clique-Struktur im frühen Fenster.
|
||||
\feld{ETH} $n$ und $T$ linear in $|V| + |E|$; mit Clique-Schranke ($|E|$-Version, zusammenhängend): nicht in $2^{o(\sqrt{n})}$, $2^{o(\sqrt{T})}$.
|
||||
|
||||
\pblock{ILP-FEASIBILITY}
|
||||
\feld{Def} $A x \le b$, $x \in \mathbb{Z}^N_{\ge 0}$ lösbar?
|
||||
\feld{Reduktion} von 3-\prob{Sat} (SS23-Klausur).
|
||||
\feld{Konstruktion} Variable je Literal ($N = 2n$); pro Klausel $-\sum_{\ell \in C_i} x_\ell \le -1$; pro Variable $x_v + x_{\bar v} \le 1$ und $-x_v - x_{\bar v} \le -1$ ($M = m + 2n$).
|
||||
\hin Belegung $\to$ 0/1-Vektor erfüllt alles.
|
||||
\rueck Paar-Gleichungen erzwingen $x_v + x_{\bar v} = 1$ (0/1-Belegung); Klausel-Zeile = erfülltes Literal.
|
||||
\feld{ETH} $M = O(m)$, $N = O(n)$ $\Rightarrow$ nicht in $2^{o(M)}$, $2^{o(N)}$.
|
||||
|
||||
\pblock{HALT$_{\mathrm{TM}}$ (nur NP-schwer!)}
|
||||
\feld{Def} Hält DTM $M$ auf Wort $w$?
|
||||
\feld{Reduktion} von 3-\prob{Sat} (HA 11.2).
|
||||
\feld{Konstruktion} $M_\varphi$: probiere alle $2^n$ Belegungen; falls eine erfüllt, halte; sonst Endlosschleife. $w = \langle \varphi \rangle$.
|
||||
\hin Erfüllbar $\to$ $M_\varphi$ findet Belegung und hält.
|
||||
\rueck $M_\varphi$ hält nur im Erfolgsfall $\to$ erfüllbar.
|
||||
\feld{Merke} NICHT in NP (unentscheidbar) $\to$ NP-schwer $\ne$ NP-vollständig. Reduktion muss polynomiell sein -- $M_\varphi$ wird nur \emph{konstruiert}, nicht ausgeführt!
|
||||
|
||||
\pblock{Klausur-Varianten-Tricks (Zusatzeigenschaft erzwingen)}
|
||||
\feld{Universalknoten} $k$-\prob{CliqueUniversal} (HA 10.2): Universalknoten $u$ anhängen, $k' = k + 1$. Rück-Richtung: obdA $u$ in der Clique (sonst tauschen).
|
||||
\feld{Verbotener Knoten} \prob{Clique-Nomember} (Präsenz 11.2): isolierten Knoten $v$ anhängen; isoliert $\to$ nie in einer Clique $\ge 2$.
|
||||
\feld{Clique UND IS} \prob{CliqueAndIndependentSet} (SS24): $k$ isolierte Knoten anhängen $\to$ IS der Größe $k$ existiert immer; Clique unverändert.
|
||||
\feld{Mindestgrad} $k$-\prob{Clique-Deg-3}, 3-\prob{Color} Grad 3, Hitchhiker's-\prob{HC} Grad 42: \skizze Grad per Gadget auffüllen, ohne neue Lösungen zu schaffen (Pendant-Dreiecke/Pfad-Gadgets, die keine neue Clique/Färbung/Tour ermöglichen). Bei HC: Gadget so, dass es von jedem Ham-Kreis zwangsläufig komplett durchlaufen wird.
|
||||
\feld{Feste Kardinalität} \prob{SubsetSumCardinality}: $+1$-Shift und Einser-Padding (s.o.).
|
||||
\feld{Grundmuster} Immer dieselben 5 Schritte: $\in$NP $\cdot$ Konstruktion $\cdot$ $\Rightarrow$ $\cdot$ $\Leftarrow$ (mit \emph{beliebiger} Lösung starten, obdA-Tausch) $\cdot$ Laufzeit + Schlusssatz.
|
||||
|
||||
\end{document}
|
||||
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|
||||
\documentclass[11pt]{article}
|
||||
\input{style.tex}
|
||||
|
||||
\title{\textbf{AAK -- Fundament}\\[0.4em]
|
||||
\large Komplexitätstheorie und Approximation ohne Beweise\\[0.2em]
|
||||
\normalsize Grundwissen zu Serie 9--13 und Skript Kapitel 6--7}
|
||||
\author{Klausurvorbereitung \glqq Analyse von Algorithmen und Komplexität\grqq{} (CAU Kiel)}
|
||||
\date{Sommersemester 2026}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\noindent\textbf{Wie du dieses Dokument benutzt.}
|
||||
Dieses PDF baut das Fundament auf: alle Begriffe, Probleme und Resultate des
|
||||
Klausurstoffs -- bewusst \emph{ohne} Beweise. Zu jedem Satz steht höchstens eine
|
||||
einzeilige Beweisidee. Arbeite es von vorne nach hinten durch. Wenn alles sitzt,
|
||||
nimm dir \textbf{Uebung.pdf} vor: Dort lernst du die Beweistechniken und
|
||||
trainierst die konkreten Klausur-Aufgabentypen.
|
||||
|
||||
\medskip
|
||||
\noindent\textbf{Klausurstoff:} Serien 9--13 und Skript Kapitel 6 (Komplexitätstheorie)
|
||||
und 7 (Approximative Algorithmen). Mehr kommt nicht dran.
|
||||
|
||||
\tableofcontents
|
||||
\newpage
|
||||
|
||||
% ==================================================================
|
||||
\section{Grundbegriffe: Sprachen und Entscheidungsprobleme}
|
||||
% ==================================================================
|
||||
|
||||
Die Komplexitätstheorie klassifiziert, \emph{wie schwer} Probleme sind.
|
||||
Ausgangspunkt: Für viele Probleme (z.\,B. Clique) kennt man nur Algorithmen mit
|
||||
exponentieller Laufzeit. Der naive Clique-Algorithmus (teste alle $k$-elementigen
|
||||
Teilmengen) braucht im schlimmsten Fall mindestens $2^{|V|/2}$ Schritte
|
||||
(Lemma~6.2 im Skript). Die Frage \glqq Geht es schneller?\grqq{} führt auf P, NP
|
||||
und die NP-Vollständigkeit.
|
||||
|
||||
\begin{defbox}{Alphabet, Wort, Sprache}
|
||||
\begin{itemize}
|
||||
\item Ein \emph{Alphabet} $\Sigma$ ist eine endliche Menge von Symbolen.
|
||||
\item $\Sigma^*$ ist die Menge aller endlichen Wörter über $\Sigma$; $|x|$ ist die Länge von $x$.
|
||||
\item Eine Teilmenge $L \subseteq \Sigma^*$ heißt \emph{Sprache}.
|
||||
\end{itemize}
|
||||
\end{defbox}
|
||||
|
||||
\begin{defbox}{Entscheidungsproblem}
|
||||
Ein \emph{Entscheidungsproblem} ist ein Tripel $(L, U, \Sigma)$ mit Sprachen
|
||||
$L \subseteq U \subseteq \Sigma^*$. Ein Algorithmus $A$ \emph{entscheidet} $L$
|
||||
(bzgl. $U$), wenn für alle $x \in U$ gilt:
|
||||
$x \in L \Rightarrow A(x) = 1$ und $x \notin L \Rightarrow A(x) = 0$.
|
||||
Meist ist $U = \Sigma^*$. Die Antwort ist immer \emph{Ja oder Nein}.
|
||||
\end{defbox}
|
||||
|
||||
Jedes konkrete Problem wird als Sprache kodiert. Beispiel Cliquenproblem:
|
||||
\begin{align*}
|
||||
\problem{Clique} = \{\, u\#v \mid\; & u \text{ kodiert Adjazenzmatrix von } G,\ v \text{ kodiert } k,\\
|
||||
& G \text{ hat Clique mit } k \text{ Knoten} \,\}.
|
||||
\end{align*}
|
||||
Dabei hat $u$ Länge $O(|V|^2)$ und $v$ Länge $O(\log |V|)$ -- Zahlen werden
|
||||
\emph{binär} kodiert, sind also exponentiell größer als ihre Kodierungslänge.
|
||||
Das wird bei SubSet Sum wichtig.
|
||||
|
||||
\begin{defbox}{Laufzeit}
|
||||
$T_A(x)$ ist die Anzahl der Operationen von Algorithmus $A$ auf Eingabe $x$.
|
||||
Die Worst-Case-Laufzeit für Eingabelänge $n$ ist
|
||||
$T_A(n) = \max\{\,T_A(x) \mid x \in \Sigma^*,\ |x| = n\,\}$.
|
||||
\end{defbox}
|
||||
|
||||
\begin{merke}
|
||||
Klausuraufgaben arbeiten fast immer mit \emph{Entscheidungsvarianten}:
|
||||
Aus einem Optimierungsproblem (\glqq finde minimales Vertex Cover\grqq) wird
|
||||
eine Ja/Nein-Frage mit Schranke (\glqq gibt es ein Vertex Cover mit $\le k$
|
||||
Knoten?\grqq).
|
||||
\end{merke}
|
||||
|
||||
% ==================================================================
|
||||
\section{Die Klassen P und NP}
|
||||
% ==================================================================
|
||||
|
||||
\subsection{P: effizient lösbar}
|
||||
|
||||
\begin{defbox}{Klasse P}
|
||||
$\Pclass$ ist die Menge aller Sprachen, die ein (deterministischer) Algorithmus
|
||||
in polynomieller Zeit entscheidet:
|
||||
\[
|
||||
\Pclass = \{\, L \subseteq \Sigma^* \mid \exists\, \text{Algorithmus } A,\;
|
||||
T_A(n) = O(n^d) \text{ mit } d = O(1),\; A \text{ entscheidet } L \,\}.
|
||||
\]
|
||||
\end{defbox}
|
||||
|
||||
Intuition: $\Pclass$ = \glqq effizient lösbar\grqq. Beispiele: Sortieren,
|
||||
kürzeste Wege, minimale Spannbäume, Matching.
|
||||
|
||||
\subsection{NP: effizient überprüfbar}
|
||||
|
||||
Für NP gibt es \textbf{zwei gleichwertige Definitionen}. Beide musst du kennen --
|
||||
die Klausur SS23 hat genau diesen Dreischritt (Zertifikat, Verifizierer, NTM)
|
||||
abgefragt.
|
||||
|
||||
\medskip\noindent\textbf{Weg 1: Verifizierer und Zertifikat.}\par
|
||||
|
||||
\begin{defbox}{Verifizierer, Zertifikat}
|
||||
Ein Algorithmus $A$ auf $\Sigma^* \times \Sigma^*$ heißt \emph{Verifizierer}
|
||||
für $L$, wenn
|
||||
\[
|
||||
L = \{\, x \in \Sigma^* \mid \exists\, c \in \Sigma^* \text{ mit } A(x,c) = 1 \,\}.
|
||||
\]
|
||||
Ein Wort $c$ mit $A(x,c) = 1$ heißt \emph{Zertifikat} für $x \in L$.
|
||||
$A$ ist ein \emph{polynomieller} Verifizierer, wenn es für jedes $x \in L$ ein
|
||||
Zertifikat $c$ gibt mit $T_A(x,c) = O(|x|^d)$ für eine Konstante $d$.
|
||||
\end{defbox}
|
||||
|
||||
\begin{defbox}{Klasse NP}
|
||||
$\NP = \{\, L \subseteq \Sigma^* \mid \text{es existiert ein polynomieller
|
||||
Verifizierer für } L \,\}$.
|
||||
\end{defbox}
|
||||
|
||||
Intuition: Eine Lösung zu \emph{finden} mag schwer sein -- eine vorgelegte
|
||||
Lösung zu \emph{prüfen} ist leicht. Das Zertifikat ist die vorgelegte Lösung.
|
||||
|
||||
\begin{bsp}{Typische Zertifikate}
|
||||
\begin{center}
|
||||
\begin{tabular}{lll}
|
||||
\toprule
|
||||
Problem & Zertifikat & Prüfung \\
|
||||
\midrule
|
||||
$k$-\problem{Clique} & Knotenmenge $C \subseteq V$ & alle Paare in $E$? $|C| \ge k$? \\
|
||||
\problem{SAT} & Belegung $\psi$ & wertet $\alpha$ zu true aus? \\
|
||||
\problem{Vertex Cover} & Knotenmenge $C$ & jede Kante überdeckt? $|C| \le k$? \\
|
||||
\problem{SubSet Sum} & Teilmenge $S$ & $\sum_{j \in S} c_j = K$? \\
|
||||
\problem{Hamiltonkreis} & Permutation $\pi$ & alle Folgekanten in $E$? \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Das Zertifikat ist polynomiell lang, die Prüfung läuft in Polynomialzeit.
|
||||
\end{bsp}
|
||||
|
||||
\medskip\noindent\textbf{Weg 2: Nichtdeterminismus.}\par
|
||||
Ein \emph{nichtdeterministischer Algorithmus} darf in jedem Schritt
|
||||
\glqq raten\grqq{} (mehrere Rechenwege). Er entscheidet $L$, wenn
|
||||
(i) jedes $w \in L$ mindestens einen akzeptierenden Rechenweg hat,
|
||||
(ii) jeder akzeptierende Rechenweg polynomiell lang ist und
|
||||
(iii) für $w \notin L$ \emph{jeder} Rechenweg ablehnt.
|
||||
Muster: \emph{Rate die Lösung, prüfe sie deterministisch.}
|
||||
|
||||
\begin{defbox}{Nichtdeterministische Turingmaschine (NDTM)}
|
||||
$M = (Q, \Gamma, q_0, \delta, F)$ mit endlicher Zustandsmenge $Q$,
|
||||
Arbeitsalphabet $\Gamma \supset \Sigma$, Startzustand $q_0$, Endzuständen
|
||||
$F \subset Q$ und Übergangsfunktion
|
||||
\[
|
||||
\delta:\; Q \times \Gamma \;\longrightarrow\; 2^{\,Q \times \Gamma \times \{-1,0,1\}}.
|
||||
\]
|
||||
Zu $(q,a)$ kann es also \emph{mehrere} (oder keine) Übergänge geben.
|
||||
Eine \emph{Konfiguration} $uqav$ speichert Bandinschrift, Zustand $q$ und
|
||||
Arbeitsfeld $a$; $C \vdash C'$ bezeichnet den Übergang zur Folgekonfiguration.
|
||||
Eine Berechnung $C_0, \dots, C_k$ \emph{akzeptiert}, wenn $C_k$ eine
|
||||
Stopkonfiguration mit Zustand in $F$ ist.
|
||||
$L \in \NP \iff$ eine NDTM akzeptiert $L$ in nichtdeterministischer
|
||||
polynomieller Zeit. Ist $|\delta(q,a)| \le 1$ überall, heißt $M$
|
||||
\emph{deterministisch} (DTM); $L \in \Pclass \iff$ eine DTM entscheidet $L$ in
|
||||
polynomieller Zeit.
|
||||
\end{defbox}
|
||||
|
||||
\begin{satzbox}{Äquivalenz der NP-Definitionen}
|
||||
Die Definition über Verifizierer und die über NDTMs beschreiben dieselbe
|
||||
Klasse $\NP$.\\[0.3em]
|
||||
\emph{Beweisidee:} Die Folge der Rateschritte einer NDTM \emph{ist} ein
|
||||
Zertifikat; umgekehrt rät die NDTM das Zertifikat (Phase 1) und simuliert den
|
||||
Verifizierer (Phase 2).
|
||||
\end{satzbox}
|
||||
|
||||
\begin{satzbox}{$\Pclass \subseteq \NP$}
|
||||
Jede Sprache in $\Pclass$ liegt auch in $\NP$.\\[0.3em]
|
||||
\emph{Beweisidee:} Der entscheidende Algorithmus ist ein Verifizierer, der das
|
||||
Zertifikat einfach ignoriert.
|
||||
\end{satzbox}
|
||||
|
||||
\begin{merke}
|
||||
Ob $\Pclass = \NP$ gilt, ist \textbf{offen} -- eines der sieben
|
||||
Millennium-Probleme (Clay Institute, 1~Mio.~Dollar). Alles Weitere
|
||||
(NP-Vollständigkeit, ETH) ist der Umgang mit dieser Unwissenheit.
|
||||
\end{merke}
|
||||
|
||||
% ==================================================================
|
||||
\section{Reduktionen, NP-Schwere, NP-Vollständigkeit}
|
||||
% ==================================================================
|
||||
|
||||
\subsection{Polynomielle Reduktionen}
|
||||
|
||||
\begin{defbox}{Polynomielle Reduktion $L_1 \redp L_2$}
|
||||
Seien $L_1 \subseteq \Sigma_1^*$, $L_2 \subseteq \Sigma_2^*$.
|
||||
Eine Funktion $f: \Sigma_1^* \to \Sigma_2^*$ mit
|
||||
\[
|
||||
w \in L_1 \iff f(w) \in L_2 \qquad \forall\, w \in \Sigma_1^*
|
||||
\]
|
||||
heißt \emph{Transformation} von $L_1$ auf $L_2$. Sie heißt \emph{polynomiell},
|
||||
wenn ein Algorithmus $f(w)$ in polynomieller Zeit berechnet (dann ist auch
|
||||
$|f(w)| \le \poly(|w|)$). Notation: $L_1 \redp L_2$
|
||||
(\glqq $L_1$ ist polynomiell reduzierbar auf $L_2$\grqq).
|
||||
\end{defbox}
|
||||
|
||||
Intuition: $L_1 \redp L_2$ heißt \emph{\glqq $L_2$ ist mindestens so schwer wie
|
||||
$L_1$\grqq} -- wer $L_2$ lösen kann, kann damit auch $L_1$ lösen.
|
||||
|
||||
\begin{warnung}
|
||||
\textbf{Die Richtung der Reduktion} ist der häufigste Fehler.
|
||||
Um zu zeigen, dass ein \emph{neues} Problem $Y$ schwer ist, reduzierst du ein
|
||||
\emph{bekanntes} schweres Problem $X$ \emph{auf} $Y$: also $X \redp Y$
|
||||
(\glqq bekannt $\redp$ neu\grqq). Die Reduktion nimmt eine $X$-Instanz und baut
|
||||
daraus eine $Y$-Instanz -- niemals umgekehrt.
|
||||
\end{warnung}
|
||||
|
||||
\begin{satzbox}{Transitivität}
|
||||
$L_1 \redp L_2$ und $L_2 \redp L_3$ $\Rightarrow$ $L_1 \redp L_3$.\\[0.3em]
|
||||
\emph{Beweisidee:} Reduktionen hintereinanderschalten; die Ausgabe der ersten
|
||||
ist polynomiell groß, also bleibt die Komposition polynomiell.
|
||||
\end{satzbox}
|
||||
|
||||
\subsection{NP-schwer und NP-vollständig}
|
||||
|
||||
\begin{defbox}{NP-schwer, NP-vollständig}
|
||||
\begin{itemize}
|
||||
\item $L_0$ heißt \emph{NP-schwer} $\iff$ $\forall L \in \NP:\; L \redp L_0$.
|
||||
\item $L_0$ heißt \emph{NP-vollständig} $\iff$ $L_0 \in \NP$ \textbf{und} $L_0$ ist NP-schwer.
|
||||
\end{itemize}
|
||||
\end{defbox}
|
||||
|
||||
\begin{merke}
|
||||
Eselsbrücke: \emph{NP-schwer} = untere Schranke (mindestens so schwer wie alles
|
||||
in NP). \emph{$\in \NP$} = obere Schranke (nicht schwerer als NP).
|
||||
\emph{NP-vollständig} = beides. Es gibt NP-schwere Probleme, die nicht in NP
|
||||
liegen -- z.\,B. das Halteproblem (Abschnitt~\ref{sec:halt}).
|
||||
\end{merke}
|
||||
|
||||
\begin{satzbox}{NP-vollständig und P=NP}
|
||||
Sei $L_0$ NP-vollständig. Dann gilt: $\Pclass = \NP \iff L_0 \in \Pclass$.\\[0.3em]
|
||||
\emph{Beweisidee ($\Leftarrow$):} Für beliebiges $L \in \NP$ schalte die
|
||||
Reduktion $L \redp L_0$ vor den Polynomialzeit-Algorithmus für $L_0$.
|
||||
\end{satzbox}
|
||||
|
||||
Konsequenz: Findet jemand für \emph{ein} NP-vollständiges Problem einen
|
||||
Polynomialzeit-Algorithmus, kollabiert alles: $\Pclass = \NP$. Solange das
|
||||
nicht passiert, gilt ein NP-Vollständigkeitsbeweis als starkes Indiz, dass es
|
||||
keinen effizienten Algorithmus gibt.
|
||||
|
||||
\begin{satzbox}{Vererbungskorollar -- das wichtigste Werkzeug}
|
||||
Ist $L_0$ NP-vollständig, $L_0 \redp L_1$ und $L_1 \in \NP$, so ist $L_1$
|
||||
NP-vollständig.\\[0.3em]
|
||||
\emph{Beweisidee:} $L \redp L_0 \redp L_1$ für alle $L \in \NP$ (Transitivität).
|
||||
\end{satzbox}
|
||||
|
||||
Dieses Korollar ist das Standardwerkzeug der Klausur: Statt \glqq alle $L \in
|
||||
\NP$\grqq{} zu betrachten, reicht \emph{eine} Reduktion von einem bekannten
|
||||
NP-vollständigen Problem plus der Nachweis $L_1 \in \NP$.
|
||||
|
||||
\subsection{Der Startpunkt: Satz von Cook--Levin}
|
||||
|
||||
\begin{satzbox}{Cook (1971) / Levin (1973)}
|
||||
\problem{SAT} ist NP-vollständig.\\[0.3em]
|
||||
\emph{Beweisidee:} Für beliebiges $L \in \NP$ mit NDTM $M$ wird die Berechnung
|
||||
von $M$ auf $u$ als KNF-Formel $\alpha_u$ kodiert:
|
||||
$\alpha_u = \alpha_{\text{Anfang}} \wedge \alpha_{\text{Ende}} \wedge
|
||||
\alpha_{\text{Eindeutig}} \wedge \alpha_{\text{Übergang}}$ mit Variablen für
|
||||
\glqq Zustand zur Zeit $t$\grqq, \glqq Arbeitsfeld zur Zeit $t$\grqq,
|
||||
\glqq Bandinhalt Feld $i$ zur Zeit $t$\grqq. Erfüllende Belegungen entsprechen
|
||||
genau akzeptierenden Berechnungen; die Formel hat polynomielle Größe
|
||||
$O(T(n)^3 \log T(n))$.
|
||||
\end{satzbox}
|
||||
|
||||
Ab hier werden alle weiteren NP-Vollständigkeitsbeweise per Reduktionskette aus
|
||||
SAT aufgebaut (Kapitel~\ref{sec:landkarte}).
|
||||
|
||||
% ==================================================================
|
||||
\section{Problemkatalog}\label{sec:katalog}
|
||||
% ==================================================================
|
||||
|
||||
Alle Probleme des Klausurstoffs im Steckbrief-Format. Präge dir zu jedem ein:
|
||||
Eingabe, Frage, Status, und \emph{woher} die NP-Vollständigkeit kommt.
|
||||
|
||||
\subsection{Logik-Probleme}
|
||||
|
||||
\begin{problembox}{SAT}
|
||||
\textbf{Gegeben:} Boolescher Ausdruck $\alpha = C_1 \wedge \dots \wedge C_m$ in
|
||||
konjunktiver Normalform (KNF): Klauseln $C_i$ sind Disjunktionen von Literalen
|
||||
($x_j$ oder $\neg x_j$).\\
|
||||
\textbf{Entscheide:} Gibt es eine erfüllende Belegung der Variablen?\\
|
||||
\textbf{Status:} NP-vollständig (Cook--Levin). \emph{Das} Ur-Problem.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{3-SAT}
|
||||
\textbf{Gegeben:} KNF-Formel mit \emph{höchstens 3 Literalen pro Klausel}.\\
|
||||
\textbf{Entscheide:} Erfüllbar?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{SAT} \redp \problem{3-SAT}$
|
||||
(lange Klauseln mit Hilfsvariablen aufspalten).
|
||||
Startpunkt fast aller ETH-Reduktionen. Variante \problem{3-SAT$'$}
|
||||
(\emph{genau} 3 Literale) ist ebenfalls NP-vollständig.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{MAX-3-SAT (Optimierungsproblem)}
|
||||
\textbf{Gegeben:} KNF-Formel, jede Klausel mit 3 Literalen.\\
|
||||
\textbf{Gesucht:} Belegung $\beta$, die die Anzahl $v(\beta)$ erfüllter
|
||||
Klauseln maximiert.\\
|
||||
\textbf{Status:} Optimierungsvariante von 3-SAT; in der Klausur als
|
||||
\emph{Approximationsaufgabe} (Güte-2-Algorithmus, siehe
|
||||
Abschnitt~\ref{sec:approx-max3sat}).
|
||||
\end{problembox}
|
||||
|
||||
\subsection{Graphprobleme}
|
||||
|
||||
\begin{problembox}{$k$-Clique}
|
||||
\textbf{Gegeben:} Ungerichteter Graph $G = (V,E)$, Zahl $k \ge 1$.
|
||||
Eine \emph{Clique} ist $C \subseteq V$ mit $\{u,v\} \in E$ für alle $u \ne v$
|
||||
aus $C$.\\
|
||||
\textbf{Entscheide:} Existiert eine Clique mit $|C| \ge k$?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{SAT} \redp \problem{$k$-Clique}$
|
||||
(Vorlesungsbeweis -- auswendig können!).
|
||||
\end{problembox}
|
||||
|
||||
\begin{bsp}{Clique}
|
||||
\setlength{\unitlength}{0.9cm}
|
||||
\begin{center}
|
||||
\begin{picture}(3.6,2.4)(-0.3,-0.2)
|
||||
\put(0,1){\circle*{0.12}}\put(-0.3,1){\small $1$}
|
||||
\put(1,2){\circle*{0.12}}\put(1,2.15){\small $2$}
|
||||
\put(2,1){\circle*{0.12}}\put(2.1,1.15){\small $3$}
|
||||
\put(1.5,0){\circle*{0.12}}\put(1.4,-0.35){\small $4$}
|
||||
\put(3,0){\circle*{0.12}}\put(3.1,-0.2){\small $5$}
|
||||
\put(0,1){\line(1,1){1}} % 1-2
|
||||
\put(0,1){\line(1,0){2}} % 1-3
|
||||
\put(1,2){\line(1,-1){1}} % 2-3
|
||||
\put(2,1){\line(-1,-2){0.5}} % 3-4
|
||||
\put(2,1){\line(1,-1){1}} % 3-5
|
||||
\end{picture}
|
||||
\end{center}
|
||||
$C = \{1,2,3\}$ ist eine Clique der Größe 3; $\{2,3,4\}$ nicht (Kante
|
||||
$\{2,4\}$ fehlt).
|
||||
\end{bsp}
|
||||
|
||||
\begin{problembox}{Independent Set (IS)}
|
||||
\textbf{Gegeben:} $G = (V,E)$, Zahl $k$. Ein \emph{Independent Set} ist
|
||||
$I \subseteq V$ ohne Kante zwischen zwei Knoten aus $I$.\\
|
||||
\textbf{Entscheide:} Gibt es ein Independent Set mit $|I| \ge k$?\\
|
||||
\textbf{Status:} NP-vollständig -- $C$ ist Clique in $G$ $\iff$ $C$ ist
|
||||
Independent Set im Komplementgraphen $\bar{G}$.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Vertex Cover (VC)}
|
||||
\textbf{Gegeben:} $G = (V,E)$, Zahl $k$. Ein \emph{Vertex Cover} ist
|
||||
$C \subseteq V$, sodass jede Kante mindestens einen Endpunkt in $C$ hat.\\
|
||||
\textbf{Entscheide:} Gibt es ein Vertex Cover mit $|C| \le k$?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{Clique} \redp \problem{VC}$.
|
||||
\end{problembox}
|
||||
|
||||
\begin{merke}
|
||||
\textbf{Dualitätsdreieck} (Kern vieler Aufgaben): In $G$ mit $n$ Knoten gilt
|
||||
\begin{align*}
|
||||
C \text{ ist Clique der Größe } k \text{ in } G
|
||||
&\iff C \text{ ist IS der Größe } k \text{ in } \bar{G}\\
|
||||
&\iff V \setminus C \text{ ist VC der Größe } n-k \text{ in } \bar{G}.
|
||||
\end{align*}
|
||||
Merksatz: \emph{Vertex Cover = Komplement eines Independent Set}
|
||||
(in demselben Graphen), \emph{Clique = Independent Set im Komplementgraphen}.
|
||||
\end{merke}
|
||||
|
||||
\begin{problembox}{$k$-Color (Färbungsproblem)}
|
||||
\textbf{Gegeben:} $G = (V,E)$, Zahl $k$. Eine \emph{$k$-Färbung} ist
|
||||
$f: V \to \{1,\dots,k\}$ mit $f(u) \ne f(v)$ für alle $\{u,v\} \in E$.\\
|
||||
\textbf{Entscheide:} Ist $G$ $k$-färbbar?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{3-SAT} \redp \problem{$k$-Color}$
|
||||
(Konstruktion mit $n+1$ Farben, Knoten $x_i, \bar{x}_i, v_i$, Klauselknoten
|
||||
$F_j$ und Zentrum $z$). Auch \problem{3-Color} ist NP-vollständig.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Hamiltonkreis (HK) und Hamiltonpfad}
|
||||
\textbf{Gegeben:} $G = (V,E)$ ungerichtet (Aussage gilt auch gerichtet).\\
|
||||
\textbf{Entscheide:} Gibt es einen Kreis, der jeden Knoten \emph{genau einmal}
|
||||
besucht (formal: Permutation $\pi$ mit $\{v_{\pi(i)}, v_{\pi(i+1)}\} \in E$
|
||||
zyklisch)?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{3-SAT}' \redp \problem{HK}$
|
||||
(Gadget-Konstruktion mit A-/B-Komponenten; nur Beweisskizze im Skript).
|
||||
\problem{Hamiltonpfad} (Pfad statt Kreis) ist ebenfalls NP-vollständig;
|
||||
Klausur SS23 verlangte die Reduktionen zwischen beiden in \emph{beide}
|
||||
Richtungen.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{TSP (Traveling Salesman, Entscheidungsvariante)}
|
||||
\textbf{Gegeben:} Knoten $\{1,\dots,n\}$, Distanzen $d(i,j)$, Schranke $L$.\\
|
||||
\textbf{Entscheide:} Gibt es eine Rundreise (jeder Knoten genau einmal) mit
|
||||
Länge $\le L$?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{HK} \redp \problem{TSP}$
|
||||
(Kanten $\mapsto$ Distanz 1, Nicht-Kanten $\mapsto |V|+1$, $L = |V|$).
|
||||
Bleibt NP-vollständig im metrischen Fall ($\Delta$-Ungleichung).
|
||||
Als Optimierungsproblem zentral in Kapitel~\ref{sec:approx}.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Feedback Vertex Set (FVS)}
|
||||
\textbf{Gegeben:} \emph{Gerichteter} Graph $G = (V,E)$, Zahl $k$.\\
|
||||
\textbf{Entscheide:} Gibt es $X \subseteq V$ mit $|X| \le k$, sodass
|
||||
$G \setminus X$ kreisfrei ist?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{VC} \redp \problem{FVS}$
|
||||
(Serie 9/10: jede ungerichtete Kante wird zu zwei antiparallelen Bögen).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{$\Delta$-Cover (Dreiecksüberdeckung)}
|
||||
\textbf{Gegeben:} $G = (V,E)$ ungerichtet, Zahl $k$.\\
|
||||
\textbf{Entscheide:} Gibt es $C_\Delta \subseteq V$, $|C_\Delta| \le k$, das
|
||||
\emph{jede 3-Clique} (Dreieck) von $G$ trifft?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{VC} \redp \Delta\problem{-Cover}$
|
||||
(Serie 11: jede Kante zu einem Dreieck aufblasen).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Dominating Set}
|
||||
\textbf{Gegeben:} $G = (V,E)$, Zahl $k$.\\
|
||||
\textbf{Entscheide:} Gibt es $D \subseteq V$, $|D| \le k$, sodass jeder Knoten
|
||||
in $D$ liegt oder einen Nachbarn in $D$ hat?\\
|
||||
\textbf{Status:} NP-vollständig. In Klausuren als NP-Zugehörigkeits- und
|
||||
ETH-Beispiel (Reduktion von 3-SAT vorgegeben).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Longest Path}
|
||||
\textbf{Gegeben:} $G = (V,E)$, Zahl $k$.\\
|
||||
\textbf{Entscheide:} Gibt es einen einfachen Pfad der Länge $\ge k$?\\
|
||||
\textbf{Status:} NP-vollständig (verallgemeinert Hamiltonpfad). In der Klausur
|
||||
SS23 als NP-Zugehörigkeits-Aufgabe (Zertifikat/Verifizierer/NTM).
|
||||
\end{problembox}
|
||||
|
||||
\subsection{Mengen- und Zahlprobleme}
|
||||
|
||||
\begin{problembox}{3-dimensionales Matching (3-DM)}
|
||||
\textbf{Gegeben:} Mengen $U, V, W$ mit $|U| = |V| = |W|$ und
|
||||
$T \subseteq V \times W \times U$.\\
|
||||
\textbf{Entscheide:} Gibt es $M \subseteq T$ mit $|M| = |U|$, sodass keine zwei
|
||||
Tripel in einer Komponente übereinstimmen?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{SAT} \redp \problem{3-DM}$
|
||||
(Gadgets: $a/b$-Knoten für Variablensetzung, $v/w$ für Klauselwahl, $c/d$ als
|
||||
Garbage Collection).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{3-Exact Cover (3-EC)}
|
||||
\textbf{Gegeben:} Familie $F = \{S_1, \dots, S_n\}$ von 3-elementigen
|
||||
Teilmengen einer Menge $U$ mit $|U| = 3m$.\\
|
||||
\textbf{Entscheide:} Gibt es $m$ Mengen aus $F$, die $U$ exakt überdecken
|
||||
(disjunkt, Vereinigung $= U$)?\\
|
||||
\textbf{Status:} NP-vollständig -- jede 3-DM-Instanz \emph{ist} eine
|
||||
3-EC-Instanz (Uminterpretation).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{SubSet Sum}
|
||||
\textbf{Gegeben:} Ganze Zahlen $c_1, \dots, c_n$ und Zielwert $K$.\\
|
||||
\textbf{Entscheide:} Gibt es $S \subseteq \{1,\dots,n\}$ mit
|
||||
$\sum_{j \in S} c_j = K$?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{3-EC} \redp \problem{SubSet Sum}$
|
||||
(Mengen als Bitvektoren zur Basis $n+1$).
|
||||
\textbf{Klausur-Spitzenreiter:} In jeder Altklausur taucht eine
|
||||
SubSet-Sum-Variante auf.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Partition}
|
||||
\textbf{Gegeben:} Ganze Zahlen $c_1, \dots, c_n$.\\
|
||||
\textbf{Entscheide:} Gibt es $S$ mit
|
||||
$\sum_{j \in S} c_j = \tfrac{1}{2}\sum_{j=1}^n c_j$?\\
|
||||
\textbf{Status:} NP-vollständig, via $\problem{SubSet Sum} \redp
|
||||
\problem{Partition}$ (zwei Zusatzzahlen $N - K$ und $K + 1$).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Rucksackproblem (Knapsack, Entscheidungsvariante)}
|
||||
\textbf{Gegeben:} $n$ Gegenstände mit Größen $c_j$ und Profiten $p_j$,
|
||||
Kapazität $K$, Zielprofit $P$.\\
|
||||
\textbf{Entscheide:} Gibt es $S$ mit $\sum_{j \in S} c_j \le K$ und
|
||||
$\sum_{j \in S} p_j \ge P$?\\
|
||||
\textbf{Status:} NP-vollständig -- SubSet Sum ist der Spezialfall $c_j = p_j$,
|
||||
$P = K$. Als Optimierungsproblem zentral in Kapitel~\ref{sec:approx}
|
||||
(Greedy, FPTAS).
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Hitting Set}
|
||||
\textbf{Gegeben:} Universum $U$, Teilmengen $F_1, \dots, F_r \subseteq U$,
|
||||
Zahl $k$.\\
|
||||
\textbf{Entscheide:} Gibt es $S \subseteq U$, $|S| \le k$, mit
|
||||
$S \cap F_i \ne \emptyset$ für alle $i$ (\glqq $S$ trifft jede Menge\grqq)?\\
|
||||
\textbf{Status:} NP-vollständig; via $\problem{3-SAT} \redp \problem{Hitting
|
||||
Set}$ (Serie 13). Standard-Beispiel für ETH-Lower-Bounds.
|
||||
\end{problembox}
|
||||
|
||||
\begin{problembox}{Set Cover}
|
||||
\textbf{Gegeben:} Universum $U$, Mengen $S_1, \dots, S_r \subseteq U$, Zahl $k$.\\
|
||||
\textbf{Entscheide:} Gibt es $\le k$ Mengen, deren Vereinigung $U$ ist?\\
|
||||
\textbf{Status:} NP-vollständig; dual zu Hitting Set (Klausur SS23:
|
||||
$\problem{Set Cover} \redp \problem{Hitting Set}$ für ETH-Schranken).
|
||||
\end{problembox}
|
||||
|
||||
\subsection{Scheduling}
|
||||
|
||||
\begin{problembox}{$P\,\|\,C_{\max}$ (Scheduling auf identischen Maschinen)}
|
||||
\textbf{Gegeben:} $n$ Jobs mit Laufzeiten $p_1, \dots, p_n$, $m$ identische
|
||||
Maschinen.\\
|
||||
\textbf{Gesucht:} Partition der Jobs in $B_1, \dots, B_m$ mit minimalem
|
||||
\emph{Makespan} $\Cmax = \max_i \sum_{J_j \in B_i} p_j$.\\
|
||||
\textbf{Status:} NP-vollständig schon für $m = 2$ (Reduktion von
|
||||
\problem{Partition}). Zentrale Approximationsalgorithmen: List Scheduling,
|
||||
LPT (Kapitel~\ref{sec:approx}).
|
||||
\end{problembox}
|
||||
|
||||
% ==================================================================
|
||||
\section{Die Reduktionslandkarte}\label{sec:landkarte}
|
||||
% ==================================================================
|
||||
|
||||
Alle NP-Vollständigkeitsbeweise des Stoffs hängen an dieser Kette
|
||||
(Pfeil $=$ \glqq wird reduziert auf\grqq):
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{ccccc}
|
||||
& & $\boxed{\textbf{\problem{SAT}}}$ \; {\footnotesize (Cook--Levin)} & & \\[0.5em]
|
||||
$\swarrow$ & & $\downarrow$ & & $\searrow$ \\[0.5em]
|
||||
$\problem{3-SAT}$ & & $\problem{$k$-Clique}$ & & $\problem{3-DM}$ \\[0.5em]
|
||||
$\downarrow$ & & $\downarrow$ & & $\downarrow$ \\[0.5em]
|
||||
$\problem{$k$-Color}$ & & $\problem{IS}$, $\problem{VC}$ & & $\problem{3-EC}$ \\[0.5em]
|
||||
& & $\downarrow$ & & $\downarrow$ \\[0.5em]
|
||||
$\problem{3-SAT}' \to \problem{HK} \to \problem{TSP}$ & & $\problem{FVS}$, $\Delta$\problem{-Cover} & & $\problem{SubSet Sum}$ \\[0.5em]
|
||||
& & & & $\downarrow$ \\[0.5em]
|
||||
& & & & $\problem{Partition} \to P\,\|\,\Cmax$;\; $\problem{Knapsack}$ \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\medskip
|
||||
Die Konstruktionsideen im Überblick (Details und Beweise in Uebung.pdf):
|
||||
|
||||
\begin{center}
|
||||
\footnotesize
|
||||
\begin{tabular}{lll}
|
||||
\toprule
|
||||
Reduktion & Kernidee der Konstruktion & Quelle \\
|
||||
\midrule
|
||||
$\problem{SAT} \redp \problem{3-SAT}$ & lange Klauseln per Hilfsvariablen-Kette aufspalten & Skript 6.25 \\
|
||||
$\problem{SAT} \redp \problem{$k$-Clique}$ & Knoten $[i,j]$ pro Literal; Kanten = konsistent, andere Klausel & Skript 6.26 \\
|
||||
$\problem{3-SAT} \redp \problem{$k$-Color}$ & $n{+}1$ Farben; Clique $\{v_1,\dots,v_n,z\}$ erzwingt Setzung & Skript 6.28 \\
|
||||
$\problem{Clique} \redp \problem{VC}$ & Komplementgraph; $C$ Clique $\iff V \setminus C$ VC, $k' = n-k$ & Präsenz 10 \\
|
||||
$\problem{VC} \redp \problem{FVS}$ & Kante $\to$ zwei antiparallele Bögen (2-Kreise) & Serie 10 \\
|
||||
$\problem{VC} \redp \Delta\problem{-Cover}$ & Kante $\{u,v\} \to$ Dreieck $\{v_e, u, v\}$ & Serie 11 \\
|
||||
$\problem{SAT} \redp \problem{3-DM}$ & Variablen-Rad, Klausel-Tripel, Garbage Collection & Skript 6.30 \\
|
||||
$\problem{3-DM} \redp \problem{3-EC}$ & Uminterpretation (Tripel = 3er-Mengen) & Skript 6.31 \\
|
||||
$\problem{3-EC} \redp \problem{SubSet Sum}$ & Mengen als Bitvektoren zur Basis $n{+}1$; kein Übertrag & Skript 6.32 \\
|
||||
$\problem{SubSet Sum} \redp \problem{Partition}$ & Zusatzzahlen $N{-}K$ und $K{+}1$ & Skript 6.33 \\
|
||||
$\problem{3-SAT}' \redp \problem{HK}$ & A-Komponenten (XOR), B-Komponenten (Klauseln) & Skript 6.42 \\
|
||||
$\problem{HK} \redp \problem{TSP}$ & Kanten Distanz 1, Nicht-Kanten $|V|{+}1$, $L = |V|$ & Skript 7.2 \\
|
||||
$\problem{Partition} \redp P2\,\|\,\Cmax$ & Zahlen = Jobs; Makespan $\tfrac12\sum c_j$? & Skript 7.23 \\
|
||||
$\problem{3-SAT} \redp \problem{SubSet Sum}$ (streng) & Ziffern zur Basis 10; $|A| = O(m)$ -- für ETH & Skript 6.40 \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
% ==================================================================
|
||||
\section{ETH: Exponentialzeit-Hypothese und untere Schranken}\label{sec:eth}
|
||||
% ==================================================================
|
||||
|
||||
NP-Vollständigkeit sagt nur \glqq vermutlich nicht polynomiell\grqq. Die ETH
|
||||
ist eine \emph{stärkere} Annahme, mit der man konkrete exponentielle
|
||||
\emph{untere Laufzeitschranken} herleitet. In \textbf{jeder} Altklausur ist die
|
||||
letzte Aufgabe eine ETH-Aufgabe.
|
||||
|
||||
\begin{defbox}{Klein-o-Notation}
|
||||
$f(x) = o(g(x)) \iff$ für \emph{jedes} $c > 0$ gilt ab einem $x_0$:
|
||||
$|f(x)| \le c \cdot |g(x)|$; äquivalent $f(x)/g(x) \to 0$.
|
||||
Beispiele: $\sqrt{n} = o(n)$, $n/\log n = o(n)$, aber $\delta n \ne o(n)$ für
|
||||
festes $\delta > 0$. Ein \glqq $2^{o(n)}$-Algorithmus\grqq{} ist also einer,
|
||||
der \emph{subexponentiell} in $n$ läuft.
|
||||
\end{defbox}
|
||||
|
||||
\begin{satzbox}{Exponentialzeit-Hypothese (ETH; Impagliazzo, Paturi, Zane 2001)}
|
||||
Es existiert ein $\delta > 0$, sodass \problem{3-SAT} mit $n$ Variablen und $m$
|
||||
Klauseln \emph{nicht} in Zeit $2^{\delta n} (n+m)^{O(1)}$ gelöst werden kann.\\[0.3em]
|
||||
Äquivalent formuliert: Es gibt keinen $2^{o(n)} \cdot \poly$-Algorithmus für
|
||||
\problem{3-SAT}. (Unbewiesene Annahme; alle bekannten 3-SAT-Algorithmen
|
||||
brauchen $c^n$, derzeit bestes $c \approx 1{,}307$.)
|
||||
\end{satzbox}
|
||||
|
||||
\begin{satzbox}{Sparsification-Lemma}
|
||||
Unter der ETH existiert $\delta' > 0$, sodass \problem{3-SAT} nicht in Zeit
|
||||
$2^{\delta' m} (n+m)^{O(1)}$ lösbar ist -- also auch kein
|
||||
$2^{o(m)}$-Algorithmus (Schranke in der \emph{Klauselzahl} $m$).
|
||||
\end{satzbox}
|
||||
|
||||
\begin{merke}
|
||||
Warum braucht man das Lemma? Die ETH spricht über $n$ (Variablen). Reduktionen
|
||||
erzeugen Instanzgrößen aber oft abhängig von $m$ (Klauseln). Es gilt zwar
|
||||
$n \le 3m$ (jede Variable komme vor), aber $m$ kann bis $O(n^3)$ groß sein --
|
||||
eine $2^{o(m)}$-Schranke folgt daher \emph{nicht} direkt aus der ETH. Das
|
||||
Sparsification-Lemma schließt genau diese Lücke.
|
||||
In Klausurlösungen: Immer wenn dein Parameter an $m$ hängt, das
|
||||
Sparsification-Lemma zitieren.
|
||||
\end{merke}
|
||||
|
||||
\begin{satzbox}{ETH-Konsequenzen (Skript 6.37--6.40)}
|
||||
Unter Annahme der ETH gibt es keinen $2^{o(n)} \cdot \poly$-Algorithmus für:
|
||||
\problem{3-Color} ($n = $ Knotenzahl; via Reduktion mit $O(n+m)$ Knoten),
|
||||
\problem{Clique}, \problem{Vertex Cover}, \problem{Independent Set},
|
||||
sowie \problem{SubSet Sum} und \problem{Partition} ($n = $ Anzahl Zahlen;
|
||||
via strenger Reduktion mit $|A| = O(m)$ Items).
|
||||
\end{satzbox}
|
||||
|
||||
Das Beweisschema (Kontraposition) lernst du in Uebung.pdf, Kapitel 5:
|
||||
\emph{Gäbe es einen $2^{o(\text{Parameter})}$-Algorithmus für das Zielproblem,
|
||||
ergäbe die Reduktion einen $2^{o(m)}$-Algorithmus für 3-SAT -- Widerspruch zum
|
||||
Sparsification-Lemma.} Entscheidend ist, die Parameter der konstruierten
|
||||
Instanz (Knoten, Kanten, $k$, \dots) durch $n$ und $m$ abzuschätzen.
|
||||
|
||||
% ==================================================================
|
||||
\section{Approximative Algorithmen}\label{sec:approx}
|
||||
% ==================================================================
|
||||
|
||||
Wenn exakte Lösungen (vermutlich) exponentielle Zeit brauchen, berechnet man in
|
||||
Polynomialzeit \emph{beweisbar gute Näherungen}.
|
||||
|
||||
\subsection{Gütebegriff und Approximationsschemata}
|
||||
|
||||
\begin{defbox}{Multiplikative (Worst-Case-)Güte}
|
||||
Sei $A$ ein Algorithmus für ein Optimierungsproblem, $\OPT(I)$ der optimale
|
||||
Wert zu Instanz $I$. $A$ hat Güte $\alpha$, wenn für \emph{alle} Instanzen $I$:
|
||||
\begin{itemize}
|
||||
\item Minimierungsproblem: $A(I) \le \alpha \cdot \OPT(I)$ \quad (z.\,B. TSP, VC, $\Cmax$),
|
||||
\item Maximierungsproblem: $A(I) \ge \tfrac{1}{\alpha} \cdot \OPT(I)$ \quad (z.\,B. Knapsack, MAX-3-SAT).
|
||||
\end{itemize}
|
||||
Die Güte ist \emph{scharf}, wenn es Instanzen gibt, die den Faktor (beliebig
|
||||
genau) erreichen.
|
||||
\end{defbox}
|
||||
|
||||
\begin{defbox}{PTAS, EPTAS, FPTAS}
|
||||
Eine Familie $(A_\varepsilon)$ von Algorithmen mit Güte $1 + \varepsilon$ für
|
||||
jedes $\varepsilon > 0$ heißt
|
||||
\begin{itemize}
|
||||
\item \emph{PTAS} (polynomielles Approximationsschema): Laufzeit polynomiell in
|
||||
$n$ für jedes feste $\varepsilon$ (z.\,B. $O(n^{1/\varepsilon})$),
|
||||
\item \emph{EPTAS}: Laufzeit $f(1/\varepsilon) \cdot \poly(n)$
|
||||
(z.\,B. $2^{O(1/\varepsilon \log^2(1/\varepsilon))} + O(n)$ für $P\,\|\,\Cmax$),
|
||||
\item \emph{FPTAS}: Laufzeit polynomiell in $n$ \emph{und} $1/\varepsilon$
|
||||
(z.\,B. $O(n^3/\varepsilon)$ für Knapsack). Stärkste Form.
|
||||
\end{itemize}
|
||||
\emph{Pseudo-polynomiell} heißt eine Laufzeit, die polynomiell in den
|
||||
\emph{Zahlenwerten} der Eingabe ist (z.\,B. $O(n^2 p_{\max})$ beim
|
||||
Knapsack-DP) -- aber exponentiell in der Kodierungslänge.
|
||||
\end{defbox}
|
||||
|
||||
\subsection{TSP: $\Delta$TSP$_1$ und Christofides}
|
||||
|
||||
\begin{satzbox}{Allgemeines TSP ist nicht approximierbar}
|
||||
Für das allgemeine TSP gibt es \emph{keinen} Approximationsalgorithmus mit
|
||||
beschränkter Güte $\alpha$, außer $\Pclass = \NP$.\\[0.3em]
|
||||
\emph{Beweisidee:} Reduktion von \problem{HK}: Nicht-Kanten bekommen Distanz
|
||||
$\infty$ (bzw. $\alpha |V| + 1$); ein $\alpha$-Approximationsalgorithmus könnte
|
||||
dann Hamiltonkreise erkennen. (Klausuraufgabe SS23!)
|
||||
\end{satzbox}
|
||||
|
||||
Deshalb: \emph{metrisches TSP} ($\Delta$-TSP) mit symmetrischen Distanzen und
|
||||
Dreiecksungleichung $d(i,j) \le d(i,k) + d(k,j)$. Bleibt NP-vollständig, ist
|
||||
aber approximierbar.
|
||||
|
||||
\begin{rezept}{Algorithmus $\Delta$TSP$_1$ (MST-Verdopplung) -- Güte 2}
|
||||
\begin{enumerate}
|
||||
\item Berechne minimalen Spannbaum $T$ von $K_n$ mit Distanzen $d$.
|
||||
\item Verdopple alle Kanten von $T$ $\to$ Multigraph $G$ (alle Grade gerade).
|
||||
\item Bestimme Eulerkreis $K$ in $G$.
|
||||
\item Kürze $K$ zu Rundreise $R$ ab (bereits besuchte Knoten überspringen).
|
||||
\end{enumerate}
|
||||
Güte-Argument (nur Idee): $w(T) \le \OPT$ (Tour minus Kante = Spannbaum),
|
||||
$d(K) = 2w(T)$, Abkürzen verlängert wegen $\Delta$-Ungleichung nicht.
|
||||
Also $d(R) \le 2\,\OPT$. Schranke ist asymptotisch scharf (Stern-Beispiel:
|
||||
$\OPT = n$, Ausgabe $2n - 2$).
|
||||
\end{rezept}
|
||||
|
||||
\begin{defbox}{Eulerkreis}
|
||||
Ein \emph{Eulerkreis} in einem Multigraphen besucht jede \emph{Kante} genau
|
||||
einmal. Existenzkriterium: $G$ zusammenhängend und \emph{jeder Knoten hat
|
||||
geraden Grad} (Königsberger Brückenproblem).
|
||||
\end{defbox}
|
||||
|
||||
\begin{bsp}{$\Delta$TSP$_1$ durchgerechnet (Skript-Beispiel 7.7)}
|
||||
$V = \{A,B,C,D,E\}$; optimale Tour $[C,B,D,E,A,C]$ mit Länge 8.
|
||||
Der MST hat Gewicht 6 (Kanten $CB$, $BA$ je 1, $CD$, $AE$ je 2).
|
||||
Verdopplung: Multigraph mit Gewicht 12.
|
||||
Eulerkreis $K = [C,A,E,A,C,B,C,D,C]$, Abkürzen liefert
|
||||
$R = [C,A,E,B,D,C]$ mit Länge 10 $\le 2 \cdot 8$.
|
||||
\end{bsp}
|
||||
|
||||
\begin{rezept}{Algorithmus $\Delta$TSP$_2$ (Christofides) -- Güte 1{,}5}
|
||||
\begin{enumerate}
|
||||
\item Berechne minimalen Spannbaum $T$.
|
||||
\item $X :=$ Knoten mit \emph{ungeradem} Grad in $T$ ($|X|$ ist gerade).
|
||||
\item Berechne ein perfektes Matching $K$ mit minimalem Gewicht auf $X$
|
||||
(geht in $O(|V|^3)$, Lawler 1976).
|
||||
\item $G := T + K$ (alle Grade gerade), Eulerkreis, abkürzen.
|
||||
\end{enumerate}
|
||||
Güte-Argument (nur Idee): Die optimale Tour zerfällt auf $X$ in zwei Matchings
|
||||
$M_1, M_2$ mit $d(M_1) + d(M_2) \le \OPT$; also $d(K) \le \OPT/2$ und
|
||||
$d(R) \le w(T) + d(K) \le 1{,}5\,\OPT$. Auch diese Schranke ist asymptotisch
|
||||
scharf (Leiter-Beispiel, Präsenzserie 12: Rate $\to 3/2 - 1/n$).
|
||||
\end{rezept}
|
||||
|
||||
\begin{warnung}
|
||||
Klausur-Klassiker: \emph{Warum braucht Christofides die
|
||||
$\Delta$-Ungleichung?} Beim Abkürzen des Eulerkreises darf der Weg nicht länger
|
||||
werden -- das garantiert nur die Dreiecksungleichung. Und: Das Matching wird
|
||||
auf den \emph{ungerad-gradigen Knoten des MST} gebildet, nicht auf allen.
|
||||
\end{warnung}
|
||||
|
||||
\subsection{Knapsack: Greedy, MGA, FPTAS}
|
||||
|
||||
\begin{rezept}{Greedy GA -- Güte unbeschränkt!}
|
||||
Sortiere nach Profitdichte $p_i/w_i$ absteigend; nimm jeden Gegenstand, der
|
||||
noch passt. Laufzeit $O(n \log n)$.\\[0.3em]
|
||||
\textbf{Aber:} Instanz $(w_0,p_0) = (1,1)$, $(w_1,p_1) = (B, B-1)$:
|
||||
GA nimmt Gegenstand 0 (Dichte 1), Gewinn 1; optimal ist $B-1$.
|
||||
Güte $\ge B - 1$, also \emph{keine} konstante Güte.
|
||||
\end{rezept}
|
||||
|
||||
\begin{rezept}{Modified Greedy MGA -- Güte 2}
|
||||
Berechne $S_1$ mit GA \emph{und} $S_2 = \{$Gegenstand mit maximalem
|
||||
Einzelprofit$\}$; gib die bessere Lösung aus.\\[0.3em]
|
||||
Güte-Argument (nur Idee): Die fraktionale Relaxierung packt nach Dichte und
|
||||
schneidet beim \emph{Split-Item} $k{+}1$ ab:
|
||||
$\OPT \le \OPT_f \le \mathrm{GA}(I) + p_{k+1} \le \mathrm{GA}(I) + p_{\max}
|
||||
\le 2 \cdot \mathrm{MGA}(I)$.
|
||||
\end{rezept}
|
||||
|
||||
\begin{satzbox}{Sahni-Algorithmus $A_k$ ($k$-Enumeration)}
|
||||
Probiere alle Teilmengen $S$ mit $|S| \le k$ als Vorplatzierung, fülle mit GA
|
||||
auf, nimm das Beste. Güte $\le 1 + 1/k$, Laufzeit $O(n^{k+1})$ -- ein PTAS.
|
||||
\end{satzbox}
|
||||
|
||||
\begin{satzbox}{FPTAS für Knapsack (Profit-Skalierung)}
|
||||
Skaliere Profite $p_i' = \lfloor p_i / K \rfloor$ mit
|
||||
$K = \frac{\varepsilon\, p_{\max}}{(1+\varepsilon)\,n}$, löse mit dem
|
||||
pseudopolynomiellen DP ($O(n^2 p'_{\max})$), gib dessen Lösung aus.
|
||||
Laufzeit $O(n^3/\varepsilon)$, Güte $1 + \varepsilon$.\\[0.3em]
|
||||
\emph{Idee:} Durch Abrunden verliert jede Lösung höchstens $K$ pro Item, also
|
||||
$\le Kn$ gesamt; wegen $\OPT \ge p_{\max}$ ist der relative Fehler $\le \varepsilon$.
|
||||
\end{satzbox}
|
||||
|
||||
\subsection{Scheduling: List Scheduling und LPT}
|
||||
|
||||
\begin{rezept}{List Scheduling -- Güte $2 - \frac{1}{m}$}
|
||||
Gehe die Jobs in Listenreihenfolge durch; weise jeden Job der Maschine mit
|
||||
aktuell \emph{kleinster Last} zu.\\[0.3em]
|
||||
Güte-Argument (nur Idee): Sei $J_k$ der letzte Job auf der vollsten Maschine.
|
||||
Bis zu seinem Start waren alle Maschinen beschäftigt:
|
||||
$\mathrm{LS}(I) - p_k \le \frac{1}{m}\sum_i p_i \le \OPT$ und $p_k \le \OPT$
|
||||
ergeben $\mathrm{LS}(I) \le (2 - \frac{1}{m})\OPT$. Die Schranke wird von
|
||||
konkreten Instanzen erreicht (Klausuraufgabe: solche Instanzen konstruieren!).
|
||||
\end{rezept}
|
||||
|
||||
\begin{rezept}{LPT Scheduling -- Güte $\frac{4}{3} - \frac{1}{3m}$}
|
||||
\emph{Longest Processing Time first}: Sortiere die Jobs absteigend
|
||||
($p_1 \ge p_2 \ge \dots$), dann List Scheduling.\\[0.3em]
|
||||
Beweis über minimales Gegenbeispiel: Dort wäre $p_n > \OPT/3$, also höchstens
|
||||
2 Jobs pro Maschine im Optimum; per Austausch-Transformationen wird das
|
||||
Optimum in einen LPT-Schedule überführt -- Widerspruch. Auch $4/3 - 1/(3m)$
|
||||
wird asymptotisch erreicht.
|
||||
\end{rezept}
|
||||
|
||||
\begin{bsp}{List Scheduling vs. LPT, $m = 2$}
|
||||
Jobs mit Zeiten $(1, 1, 2)$ in Listenreihenfolge:
|
||||
List Scheduling legt die beiden Einsen auf $M_1$ und $M_2$, die 2 dann auf
|
||||
$M_1$ $\Rightarrow \Cmax = 3$. Optimal ist $\{2\}/\{1,1\}$ mit $\OPT = 2$.
|
||||
Die Rate ist $3/2 = 2 - \frac{1}{m}$ -- die Schranke wird exakt erreicht.
|
||||
LPT sortiert zuerst: $(2,1,1)$ ergibt $\Cmax = 2 = \OPT$.
|
||||
\end{bsp}
|
||||
|
||||
\begin{satzbox}{EPTAS für $P\,\|\,\Cmax$}
|
||||
Für jedes $\varepsilon > 0$ gibt es $A_\varepsilon$ mit
|
||||
$A_\varepsilon(I) \le (1+\varepsilon)\OPT(I)$ und Laufzeit
|
||||
$2^{O(1/\varepsilon \cdot \log^2 (1/\varepsilon))} + O(n)$.
|
||||
\end{satzbox}
|
||||
|
||||
\subsection{MAX-3-SAT: Güte 2}\label{sec:approx-max3sat}
|
||||
|
||||
\begin{rezept}{Komplementäre Belegungen -- Güte 2}
|
||||
Werte die Formel unter $\beta_0$ (alle Variablen false) und $\beta_1$ (alle
|
||||
true) aus; gib die bessere Belegung zurück.\\[0.3em]
|
||||
Warum das funktioniert: Jede Klausel enthält ein positives oder ein negatives
|
||||
Literal, wird also von $\beta_0$ \emph{oder} $\beta_1$ erfüllt. Daher
|
||||
$v(\beta_0) + v(\beta_1) \ge m$, also $\max \ge m/2 \ge \OPT/2$.
|
||||
(Beweis + scharfe Instanz: Uebung.pdf -- war Klausuraufgabe SoSe 23.)
|
||||
\end{rezept}
|
||||
|
||||
% ==================================================================
|
||||
\section{Halteproblem und Unentscheidbarkeit}\label{sec:halt}
|
||||
% ==================================================================
|
||||
|
||||
\begin{problembox}{HALT$_{\text{TM}}$ (Halteproblem)}
|
||||
\textbf{Gegeben:} Kodierung $\langle M, w \rangle$ einer DTM $M$ und eines
|
||||
Wortes $w$.\\
|
||||
\textbf{Entscheide:} Hält $M$ auf $w$ (erreicht eine Stopkonfiguration)?\\
|
||||
\textbf{Status:} \emph{Unentscheidbar} -- kein Algorithmus löst es, egal mit
|
||||
welcher Laufzeit. Trotzdem \textbf{NP-schwer} (Serie 11): via
|
||||
$\problem{SAT} \redp \problem{HALT}_{\text{TM}}$; konstruiere $M_\varphi$, die
|
||||
alle Belegungen durchprobiert und genau dann hält, wenn eine erfüllende
|
||||
existiert.
|
||||
\end{problembox}
|
||||
|
||||
\begin{merke}
|
||||
HALT$_{\text{TM}}$ ist NP-schwer, aber \emph{nicht} NP-vollständig -- es liegt
|
||||
nicht in NP (nicht einmal entscheidbar). Wichtig ist die Pointe der Reduktion:
|
||||
$M_\varphi$ läuft womöglich ewig, aber die \emph{Konstruktion} von
|
||||
$\langle M_\varphi, \varepsilon \rangle$ aus $\varphi$ ist polynomiell -- nur
|
||||
darauf kommt es an. (In den Altklausuren kam das Thema nie dran; Serienstoff
|
||||
ist es trotzdem.)
|
||||
\end{merke}
|
||||
|
||||
% ==================================================================
|
||||
\section{Wahr oder falsch? Die Logik-Merkkiste}
|
||||
% ==================================================================
|
||||
|
||||
Aussagen aus den Präsenzserien 10/11 -- ideale Selbstkontrolle:
|
||||
|
||||
\begin{center}
|
||||
\small
|
||||
\begin{tabular}{p{7.9cm}cp{4.4cm}}
|
||||
\toprule
|
||||
Aussage & W/F & Begründung \\
|
||||
\midrule
|
||||
Jedes Problem, das eine NDTM in Polyzeit löst, löst auch eine DTM in Polyzeit. &
|
||||
F$^*$ & Das wäre $\Pclass = \NP$; gilt nur, falls $\Pclass = \NP$. \\
|
||||
\addlinespace
|
||||
$A$ NP-schwer $\Rightarrow A \in \Pclass$ ausgeschlossen? / $A$ NP-schwer
|
||||
$\Rightarrow A \in \NP$? & F & NP-schwer sagt nichts über Zugehörigkeit zu NP
|
||||
(Bsp. Halteproblem). $A \in \Pclass$ würde $\Pclass = \NP$ implizieren. \\
|
||||
\addlinespace
|
||||
$L \in \NP$ und $L \redp \problem{3-SAT}$ $\Rightarrow$ $L$ NP-vollständig. &
|
||||
F & Gegenbeispiel $L = \emptyset$ bzw. jedes einfache $L$: reduzierbar
|
||||
\emph{auf} 3-SAT heißt nicht schwer. \\
|
||||
\addlinespace
|
||||
$L \redp \problem{3-SAT}$ und $\problem{3-SAT} \redp L$ $\Rightarrow$ $L$
|
||||
NP-vollständig. & W & $\problem{3-SAT} \redp L$ gibt NP-Schwere;
|
||||
$L \redp \problem{3-SAT}$ gibt $L \in \NP$. \\
|
||||
\addlinespace
|
||||
$L$ NP-vollständig: $L \in \Pclass \iff \Pclass = \NP$. & W & Satz 6.16. \\
|
||||
\addlinespace
|
||||
Es ist möglich, dass $\problem{3-SAT} \in \Pclass$ und
|
||||
$\problem{Clique} \notin \Pclass$. & F & $\problem{Clique} \redp
|
||||
\problem{3-SAT}$ (3-SAT ist NP-vollständig); aus $\problem{3-SAT} \in \Pclass$
|
||||
folgt $\problem{Clique} \in \Pclass$. \\
|
||||
\addlinespace
|
||||
Wenn die ETH falsch ist, gilt $\Pclass = \NP$. & F & ETH falsch heißt nur:
|
||||
3-SAT in $2^{o(n)}$ -- das kann immer noch superpolynomiell sein
|
||||
(z.\,B. $2^{\sqrt{n}}$). Umgekehrt gilt aber: $\Pclass = \NP \Rightarrow$ ETH
|
||||
falsch. \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
{\footnotesize $^*$ genauer: nicht beweisbar wahr; äquivalent zu $\Pclass = \NP$.}
|
||||
|
||||
\begin{merke}
|
||||
\textbf{Weiter geht's mit Uebung.pdf:} Dort stehen die Schritt-für-Schritt-Rezepte
|
||||
für alle sechs Klausur-Aufgabentypen, die vier Vorlesungsbeweise in voller
|
||||
Länge, alle Reduktionen der Serien mit Musterbeweis -- und Selbsttest-Aufgaben
|
||||
aus echten Klausuren.
|
||||
\end{merke}
|
||||
|
||||
\end{document}
|
||||
52
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lernmaterial/fundament4.pdf
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lernmaterial/fundament4.tex
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lernmaterial/fundament4.toc
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|
||||
\contentsline {section}{\numberline {1}Ein Problem, das sich wehrt}{3}{section.1}%
|
||||
\contentsline {subsection}{\numberline {1.1}Ein Graph ist ein Netz aus Punkten und Linien}{3}{subsection.1.1}%
|
||||
\contentsline {subsection}{\numberline {1.2}Was ist eine Clique?}{3}{subsection.1.2}%
|
||||
\contentsline {subsection}{\numberline {1.3}Der naheliegende Algorithmus -- und warum er scheitert}{4}{subsection.1.3}%
|
||||
\contentsline {subsection}{\numberline {1.4}Die Frage, die den ganzen Kurs trägt}{5}{subsection.1.4}%
|
||||
\contentsline {section}{\numberline {2}Was heißt eigentlich \quotedblbase schnell\textquotedblleft ?}{5}{section.2}%
|
||||
\contentsline {subsection}{\numberline {2.1}Geschwindigkeit misst man in der Eingabegröße}{6}{subsection.2.1}%
|
||||
\contentsline {subsection}{\numberline {2.2}Die Klasse P}{6}{subsection.2.2}%
|
||||
\contentsline {subsection}{\numberline {2.3}Warum wir alles als Ja/Nein-Frage schreiben}{6}{subsection.2.3}%
|
||||
\contentsline {subsection}{\numberline {2.4}Ein Stolperstein, der später wichtig wird}{7}{subsection.2.4}%
|
||||
\contentsline {section}{\numberline {3}Prüfen ist leichter als Lösen}{7}{section.3}%
|
||||
\contentsline {subsection}{\numberline {3.1}Die Sudoku-Beobachtung}{7}{subsection.3.1}%
|
||||
\contentsline {subsection}{\numberline {3.2}Zertifikat und Verifizierer}{8}{subsection.3.2}%
|
||||
\contentsline {subsection}{\numberline {3.3}Der NP-Dreischritt: so beweist man \quotedblbase $\in \mathsf {NP}$\textquotedblleft }{8}{subsection.3.3}%
|
||||
\contentsline {subsection}{\numberline {3.4}Die zweite Sicht: Raten}{9}{subsection.3.4}%
|
||||
\contentsline {subsection}{\numberline {3.5}P steckt in NP -- und die Millionenfrage}{10}{subsection.3.5}%
|
||||
\contentsline {section}{\numberline {4}Reduktionen: Probleme miteinander vergleichen}{10}{section.4}%
|
||||
\contentsline {subsection}{\numberline {4.1}Die Grundidee: ein Problem durch ein anderes lösen}{10}{subsection.4.1}%
|
||||
\contentsline {subsection}{\numberline {4.2}Die präzise Definition}{11}{subsection.4.2}%
|
||||
\contentsline {subsection}{\numberline {4.3}Die Richtung -- lies das zweimal}{11}{subsection.4.3}%
|
||||
\contentsline {subsection}{\numberline {4.4}Reduktionen lassen sich verketten}{12}{subsection.4.4}%
|
||||
\contentsline {section}{\numberline {5}Die härtesten Probleme: NP-vollständig}{12}{section.5}%
|
||||
\contentsline {subsection}{\numberline {5.1}Zwei Begriffe: NP-schwer und NP-vollständig}{13}{subsection.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.2}Warum ein Algorithmus für eines alle löst}{13}{subsection.5.2}%
|
||||
\contentsline {subsection}{\numberline {5.3}Der Anker: der Satz von Cook und Levin}{14}{subsection.5.3}%
|
||||
\contentsline {subsection}{\numberline {5.4}Das Arbeitspferd: das Vererbungskorollar}{14}{subsection.5.4}%
|
||||
\contentsline {subsection}{\numberline {5.5}Die Landkarte der Schwere}{14}{subsection.5.5}%
|
||||
\contentsline {section}{\numberline {6}Der Problem-Zoo}{15}{section.6}%
|
||||
\contentsline {subsection}{\numberline {6.1}SAT -- das Ur-Problem der Logik}{15}{subsection.6.1}%
|
||||
\contentsline {subsection}{\numberline {6.2}3-SAT -- SAT mit kurzen Klauseln}{16}{subsection.6.2}%
|
||||
\contentsline {subsection}{\numberline {6.3}Clique, Independent Set, Vertex Cover -- ein Trio}{16}{subsection.6.3}%
|
||||
\contentsline {subsection}{\numberline {6.4}$k$-Color -- die Landkarten-Färbung}{17}{subsection.6.4}%
|
||||
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|
||||
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|
||||
\contentsline {subsection}{\numberline {6.7}Die Mengen- und Zahlprobleme}{19}{subsection.6.7}%
|
||||
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|
||||
\contentsline {section}{\numberline {7}Reduktionen im Detail}{20}{section.7}%
|
||||
\contentsline {subsection}{\numberline {7.1}$\textsc {SAT} \leq _{p}\textsc {Clique}$: aus Logik wird ein Graph}{20}{subsection.7.1}%
|
||||
\contentsline {subsection}{\numberline {7.2}$\textsc {Clique} \leq _{p}\textsc {VC}$: das Komplement genügt}{22}{subsection.7.2}%
|
||||
\contentsline {subsection}{\numberline {7.3}$\textsc {VC} \leq _{p}\textsc {FVS}$: ungerichtet wird gerichtet}{22}{subsection.7.3}%
|
||||
\contentsline {subsection}{\numberline {7.4}$\textsc {3-SAT} \leq _{p}\textsc {3-Color}$: Wahrheit als Farbe}{23}{subsection.7.4}%
|
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\contentsline {subsection}{\numberline {7.5}$\textsc {3-EC} \leq _{p}\textsc {SubSet Sum}$: Mengen werden Zahlen}{23}{subsection.7.5}%
|
||||
\contentsline {subsection}{\numberline {7.6}$\textsc {SubSet Sum} \leq _{p}\textsc {Partition}$: zwei Zusatzzahlen}{24}{subsection.7.6}%
|
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\contentsline {section}{\numberline {8}Wie schwer \emph {genau}? Die ETH}{25}{section.8}%
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||||
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|
||||
\contentsline {subsection}{\numberline {8.3}Wie ein ETH-Schranken-Beweis aussieht}{26}{subsection.8.3}%
|
||||
\contentsline {subsection}{\numberline {8.4}Warum das Sparsification-Lemma nötig ist}{27}{subsection.8.4}%
|
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\contentsline {section}{\numberline {9}Damit leben: gute Näherungen}{28}{section.9}%
|
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\contentsline {subsection}{\numberline {9.1}Was heißt \quotedblbase gute Näherung\textquotedblleft ?}{28}{subsection.9.1}%
|
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\contentsline {subsection}{\numberline {9.2}TSP ist im Allgemeinen gar nicht approximierbar}{29}{subsection.9.2}%
|
||||
\contentsline {subsection}{\numberline {9.3}$\Delta $TSP$_1$: der Algorithmus als Bilderfolge}{29}{subsection.9.3}%
|
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\contentsline {subsection}{\numberline {9.4}Christofides: schlauer verdoppeln}{31}{subsection.9.4}%
|
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\contentsline {subsection}{\numberline {9.5}Knapsack: warum pure Gier scheitert}{32}{subsection.9.5}%
|
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\contentsline {subsection}{\numberline {9.6}Scheduling: einfache Regeln, sichtbar im Gantt-Bild}{33}{subsection.9.6}%
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\contentsline {subsection}{\numberline {9.7}MAX-3-SAT: Güte 2 fast geschenkt}{34}{subsection.9.7}%
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\contentsline {section}{\numberline {10}Jenseits von NP: das Halteproblem}{35}{section.10}%
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\contentsline {section}{\numberline {11}Das große Ganze}{35}{section.11}%
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\contentsline {subsection}{\numberline {11.1}Die Reise auf einen Blick}{36}{subsection.11.1}%
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\contentsline {section}{\numberline {A}Lösungen zu den Abruf-Fragen}{36}{appendix.A}%
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\contentsline {section}{\textbf {Teil A -- Die Grundlagen}}{4}{part*.2}%
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|
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57
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232
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BIN
lernmaterial/fundament6.pdf
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lernmaterial/fundament6.pdf
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2845
lernmaterial/fundament6.tex
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lernmaterial/fundament6.tex
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52
lernmaterial/fundament6.toc
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52
lernmaterial/fundament6.toc
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|
||||
\contentsline {section}{\textbf {Teil A -- Die Grundlagen}}{5}{part*.2}%
|
||||
\contentsline {section}{\numberline {1}Der Ausgangspunkt: ein widerspenstiges Problem}{5}{section.1}%
|
||||
\contentsline {section}{\numberline {2}Die Klasse P: was \quotedblbase schnell\textquotedblleft {} heißen soll}{6}{section.2}%
|
||||
\contentsline {section}{\numberline {3}Die Klasse NP: Lösungen prüfen statt finden}{7}{section.3}%
|
||||
\contentsline {subsection}{\numberline {3.1}Zertifikate und Verifizierer}{7}{subsection.3.1}%
|
||||
\contentsline {subsection}{\numberline {3.2}Dieselbe Klasse über das Raten}{7}{subsection.3.2}%
|
||||
\contentsline {subsection}{\numberline {3.3}Das Verhältnis von P und NP}{8}{subsection.3.3}%
|
||||
\contentsline {section}{\numberline {4}Reduktionen: das Vergleichswerkzeug}{8}{section.4}%
|
||||
\contentsline {section}{\numberline {5}NP-Vollständigkeit: die härtesten Probleme in NP}{9}{section.5}%
|
||||
\contentsline {subsection}{\numberline {5.1}Cook--Levin: das erste harte Problem}{10}{subsection.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.2}Der Dominoeffekt}{10}{subsection.5.2}%
|
||||
\contentsline {section}{\textbf {Teil B -- Der Katalog aller Probleme}}{11}{part*.3}%
|
||||
\contentsline {section}{\numberline {6}SAT}{11}{section.6}%
|
||||
\contentsline {section}{\numberline {7}3-SAT}{13}{section.7}%
|
||||
\contentsline {section}{\numberline {8}$k$-Clique}{15}{section.8}%
|
||||
\contentsline {section}{\numberline {9}Independent Set}{17}{section.9}%
|
||||
\contentsline {section}{\numberline {10}Vertex Cover}{18}{section.10}%
|
||||
\contentsline {section}{\numberline {11}Feedback Vertex Set}{20}{section.11}%
|
||||
\contentsline {section}{\numberline {12}$\Delta $-Cover}{21}{section.12}%
|
||||
\contentsline {section}{\numberline {13}$k$-Color}{23}{section.13}%
|
||||
\contentsline {section}{\numberline {14}Hamiltonkreis}{24}{section.14}%
|
||||
\contentsline {section}{\numberline {15}TSP}{26}{section.15}%
|
||||
\contentsline {section}{\numberline {16}3-dimensionales Matching}{27}{section.16}%
|
||||
\contentsline {section}{\numberline {17}3-Exact Cover}{28}{section.17}%
|
||||
\contentsline {section}{\numberline {18}SubSet Sum}{29}{section.18}%
|
||||
\contentsline {section}{\numberline {19}Partition}{32}{section.19}%
|
||||
\contentsline {section}{\numberline {20}Knapsack}{33}{section.20}%
|
||||
\contentsline {section}{\numberline {21}$P\|C_{\max }$ (Scheduling)}{34}{section.21}%
|
||||
\contentsline {section}{\numberline {22}Hitting Set}{35}{section.22}%
|
||||
\contentsline {section}{\numberline {23}Set Cover}{36}{section.23}%
|
||||
\contentsline {section}{\numberline {24}Dominating Set}{38}{section.24}%
|
||||
\contentsline {section}{\numberline {25}Longest Path}{39}{section.25}%
|
||||
\contentsline {section}{\numberline {26}Das Halteproblem}{39}{section.26}%
|
||||
\contentsline {section}{\textbf {Teil C -- Untere Schranken mit der ETH}}{40}{part*.64}%
|
||||
\contentsline {section}{\numberline {27}Das Handwerkszeug}{41}{section.27}%
|
||||
\contentsline {section}{\numberline {28}Die Schranken im Einzelnen}{42}{section.28}%
|
||||
\contentsline {section}{\textbf {Teil D -- Approximation}}{44}{part*.76}%
|
||||
\contentsline {section}{\numberline {29}Was eine Güte-Garantie ist}{44}{section.29}%
|
||||
\contentsline {section}{\numberline {30}TSP ohne Zusatzannahme: hoffnungslos}{45}{section.30}%
|
||||
\contentsline {section}{\numberline {31}$\Delta $TSP$_1$: Gerüst, Rundweg, Abkürzung}{45}{section.31}%
|
||||
\contentsline {section}{\numberline {32}Christofides: das Matching statt der Verdopplung}{47}{section.32}%
|
||||
\contentsline {section}{\numberline {33}Knapsack, Stufe 1: der pure Greedy}{48}{section.33}%
|
||||
\contentsline {section}{\numberline {34}Knapsack, Stufe 2: Modified Greedy}{49}{section.34}%
|
||||
\contentsline {section}{\numberline {35}Knapsack, Stufe 3: Sahni-Schema}{49}{section.35}%
|
||||
\contentsline {section}{\numberline {36}Knapsack, Stufe 4: das FPTAS}{50}{section.36}%
|
||||
\contentsline {section}{\numberline {37}Scheduling, Regel 1: List Scheduling}{50}{section.37}%
|
||||
\contentsline {section}{\numberline {38}Scheduling, Regel 2: LPT}{51}{section.38}%
|
||||
\contentsline {section}{\numberline {39}Parallel-Task-Scheduling: Jobs, die mehrere Maschinen brauchen}{51}{section.39}%
|
||||
\contentsline {section}{\numberline {40}Strip Packing: der NFDH-Algorithmus}{52}{section.40}%
|
||||
\contentsline {section}{\numberline {41}MAX-3-SAT: zwei Versuche reichen für Güte 2}{54}{section.41}%
|
||||
\contentsline {section}{\numberline {42}Vertex Cover: die Beide-Endpunkte-Regel}{55}{section.42}%
|
||||
\contentsline {section}{\numberline {43}Schlusswort}{55}{section.43}%
|
||||
2
lernmaterial/loesung_probeklausur1.aux
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2
lernmaterial/loesung_probeklausur1.aux
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|
||||
\relax
|
||||
\gdef \@abspage@last{9}
|
||||
172
lernmaterial/loesung_probeklausur1.fdb_latexmk
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172
lernmaterial/loesung_probeklausur1.fdb_latexmk
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|
||||
# Fdb version 4
|
||||
["pdflatex"] 1784492524.30939 "loesung_probeklausur1.tex" "loesung_probeklausur1.pdf" "loesung_probeklausur1" 1784492524.51768 0
|
||||
"/etc/texmf/web2c/texmf.cnf" 1779534375.33326 475 c0e671620eb5563b2130f56340a5fde8 ""
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"/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1095.tfm" 1136768653 3584 929cdff2b7a8c11bd4d49fd68cb0ae70 ""
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"/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty" 1644112042 7237 bdd120a32c8fdb4b433cf9ca2e7cd98a ""
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"/usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty" 1572645307 1057 525c2192b5febbd8c1f662c9468335bb ""
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"/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1673816307 1016 1c2b89187d12a2768764b83b4945667c ""
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"/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1601326656 19324 f4e4c6403dd0f1605fd20ed22fa79dea ""
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\documentclass[11pt,a4paper]{article}
|
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\usepackage[T1]{fontenc}
|
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\usepackage{lmodern}
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\usepackage[utf8]{inputenc}
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\usepackage[margin=2.5cm]{geometry}
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\usepackage{amsmath,amssymb}
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\usepackage{tikz}
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\usepackage{xcolor}
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\setlength{\parskip}{0.4em}
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\setlength{\parindent}{0pt}
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\emergencystretch=1.5em
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|
||||
% ---- Makros ----
|
||||
\newcommand{\problem}[1]{\textsc{#1}}
|
||||
\newcommand{\redp}{\le_p}
|
||||
\newcommand{\NP}{\textsf{NP}}
|
||||
\newcommand{\true}{\textsf{wahr}}
|
||||
\newcommand{\false}{\textsf{falsch}}
|
||||
\tikzset{knoten/.style={circle,draw,thick,minimum size=6.5mm,inner sep=1pt}}
|
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\definecolor{boxbg}{gray}{0.94}
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\newsavebox{\shadedbox}
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\newenvironment{shaded}
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{\par\smallskip\noindent\begin{lrbox}{\shadedbox}\begin{minipage}{\dimexpr\linewidth-2\fboxsep\relax}}
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{\end{minipage}\end{lrbox}\colorbox{boxbg}{\usebox{\shadedbox}}\par\smallskip}
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||||
|
||||
\newcommand{\aufgabe}[2]{\medskip\noindent{\large\textbf{Aufgabe #1}}\hfill\textbf{(#2)}\par\smallskip
|
||||
\noindent\rule{\linewidth}{0.4pt}\par\medskip}
|
||||
\newcommand{\lsg}{\par\smallskip\noindent\textbf{L\"osung.}\par\smallskip}
|
||||
|
||||
\begin{document}
|
||||
\thispagestyle{empty}
|
||||
|
||||
\begin{center}
|
||||
{\Large\textbf{Musterl\"osung -- Probeklausur 1}}\\[4pt]
|
||||
{\large Analyse von Algorithmen und Komplexit\"at, SS 2026}
|
||||
\end{center}
|
||||
\medskip
|
||||
\noindent\rule{\linewidth}{0.4pt}
|
||||
|
||||
% ============================= A1 =============================
|
||||
\newpage
|
||||
\aufgabe{1 \quad ANWENDUNG -- CHRISTOFIDES' ALGORITHMUS}{6 + 4 Punkte}
|
||||
Kantengewichte: $ab=1$, $ac=2$, $ad=3$, $bc=3$, $bd=4$, $cd=4$.
|
||||
|
||||
\lsg
|
||||
\textbf{a) Anwendung von Christofides ($\Delta$TSP2) ab Startknoten $a$.}
|
||||
|
||||
\textbf{Schritt 1 -- Minimaler Spannbaum $T$.} Kruskal (aufsteigend
|
||||
$ab\,1,\ ac\,2,\ ad\,3,\ bc\,3,\ bd\,4,\ cd\,4$): W\"ahle $ab\,(1)$, $ac\,(2)$;
|
||||
$ad\,(3)$ verbindet $d$. Die n\"achstteure Kante $bc\,(3)$ w\"urde einen Kreis
|
||||
schlie{\ss}en (\emph{No-Op: wird gepr\"uft, aber verworfen}); ebenso
|
||||
$bd\,(4)$, $cd\,(4)$. $T = \{ab, ac, ad\}$ ist ein Stern um $a$ mit Gewicht
|
||||
$1+2+3 = 6$ und eindeutig.
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[knoten] (a) at (0,2.4) {$a$};
|
||||
\node[knoten] (b) at (2.4,2.4) {$b$};
|
||||
\node[knoten] (c) at (0,0) {$c$};
|
||||
\node[knoten] (d) at (2.4,0) {$d$};
|
||||
\draw[very thick] (a) -- (b) node[fill=white,inner sep=1pt,font=\small,midway] {1};
|
||||
\draw[very thick] (a) -- (c) node[fill=white,inner sep=1pt,font=\small,midway] {2};
|
||||
\draw[very thick] (a) -- (d) node[fill=white,inner sep=1pt,font=\small,pos=0.3] {3};
|
||||
\node[font=\small] at (1.2,-0.7) {MST $T$, Gewicht $6$};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\textbf{Schritt 2 -- Knoten mit ungeradem Grad.} Grade in $T$: $\deg(a)=3$,
|
||||
$\deg(b)=\deg(c)=\deg(d)=1$. Alle vier Grade sind ungerade
|
||||
(jeder Grad wird gepr\"uft), also $X = \{a,b,c,d\}$.
|
||||
|
||||
\textbf{Schritt 3 -- Minimales perfektes Matching $M$ auf $X$.} Die drei
|
||||
m\"oglichen perfekten Matchings auf $\{a,b,c,d\}$:
|
||||
\begin{itemize}
|
||||
\item $\{ab, cd\} = 1 + 4 = 5$
|
||||
\item $\{ac, bd\} = 2 + 4 = 6$
|
||||
\item $\{ad, bc\} = 3 + 3 = 6$
|
||||
\end{itemize}
|
||||
Minimum eindeutig: $M = \{ab, cd\}$ mit Kosten $5$.
|
||||
|
||||
\textbf{Schritt 4 -- Multigraph $T + M$ und Eulerkreis.} Die Kante $ab$ kommt
|
||||
sowohl aus $T$ als auch aus $M$, ist also doppelt. Kanten: $ab$ (doppelt),
|
||||
$ac$, $ad$, $cd$. Grade: $\deg(a)=4$, $\deg(b)=2$, $\deg(c)=2$, $\deg(d)=2$ --
|
||||
alle gerade, der Multigraph ist eulersch. Eulerkreis ab $a$:
|
||||
\[ [\,a, b, a, c, d, a\,]. \]
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[knoten] (a) at (0,2.4) {$a$};
|
||||
\node[knoten] (b) at (2.4,2.4) {$b$};
|
||||
\node[knoten] (c) at (0,0) {$c$};
|
||||
\node[knoten] (d) at (2.4,0) {$d$};
|
||||
\draw[very thick,double] (a) -- (b) node[fill=white,inner sep=1pt,font=\small,midway] {$1\ (\times 2)$};
|
||||
\draw[very thick] (a) -- (c) node[fill=white,inner sep=1pt,font=\small,midway] {2};
|
||||
\draw[very thick] (a) -- (d) node[fill=white,inner sep=1pt,font=\small,pos=0.3] {3};
|
||||
\draw[very thick] (c) -- (d) node[fill=white,inner sep=1pt,font=\small,midway] {4};
|
||||
\node[font=\small] at (1.2,-0.7) {$T + M$ (eulersch)};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\textbf{Schritt 5 -- Abk\"urzen (shortcut).} Durchlaufe den Eulerkreis
|
||||
$[a,b,a,c,d,a]$ und \"uberspringe bereits besuchte Knoten:
|
||||
$a$ (neu, behalten -- \emph{No-Op}), $b$ (neu, behalten -- \emph{No-Op}),
|
||||
$a$ (\emph{bereits besucht $\Rightarrow$ \"uberspringen}), $c$ (neu), $d$ (neu),
|
||||
$a$ (Schluss). Resultierende Tour:
|
||||
\[ [\,a, b, c, d, a\,]. \]
|
||||
Tourl\"ange $= ab + bc + cd + da = 1 + 3 + 4 + 3 = 11$.
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\node[knoten] (a) at (0,2.4) {$a$};
|
||||
\node[knoten] (b) at (2.4,2.4) {$b$};
|
||||
\node[knoten] (c) at (0,0) {$c$};
|
||||
\node[knoten] (d) at (2.4,0) {$d$};
|
||||
\draw[very thick] (a) -- (b) node[fill=white,inner sep=1pt,font=\small,midway] {1};
|
||||
\draw[very thick] (b) -- (c) node[fill=white,inner sep=1pt,font=\small,pos=0.3] {3};
|
||||
\draw[very thick] (c) -- (d) node[fill=white,inner sep=1pt,font=\small,midway] {4};
|
||||
\draw[very thick] (a) -- (d) node[fill=white,inner sep=1pt,font=\small,pos=0.3] {3};
|
||||
\node[font=\small] at (1.2,-0.7) {Tour $[a,b,c,d,a]$, L\"ange $11$};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
Zum Vergleich (Brute-Force \"uber alle $3$ Touren): $[a,b,c,d,a]=11$,
|
||||
$[a,b,d,c,a]=1+4+4+2=11$, $[a,c,b,d,a]=2+3+4+3=12$. Also $\mathrm{OPT}=11$; hier
|
||||
liefert Christofides sogar das Optimum, die G\"ute $\tfrac32$ ist eingehalten
|
||||
($\tfrac{11}{11}=1 \le \tfrac32$).
|
||||
|
||||
\medskip
|
||||
\textbf{b) Bedeutung, Zusatzeigenschaften, G\"ute.}
|
||||
\begin{itemize}
|
||||
\item \textbf{Was berechnet Christofides?} Eine Rundreise (Hamiltonkreis) durch
|
||||
alle Knoten, also eine zul\"assige N\"aherungsl\"osung f\"ur das metrische
|
||||
Travelling-Salesperson-Problem.
|
||||
\item \textbf{Zusatzeigenschaft 1 -- Symmetrie:} $w(u,v) = w(v,u)$ f\"ur alle
|
||||
Kanten (die Kostenfunktion ist ungerichtet). \emph{Verletzende Eingabe:} ein
|
||||
gerichteter Graph mit $w(a,b) = 1$, aber $w(b,a) = 5$. Dann sind MST und Matching
|
||||
nicht wohldefiniert bzw.\ die Analyse bricht.
|
||||
\item \textbf{Zusatzeigenschaft 2 -- Dreiecksungleichung:} $w(u,v) \le w(u,x) +
|
||||
w(x,v)$ f\"ur alle Knoten $u,x,v$. \emph{Verletzende Eingabe:} drei Knoten
|
||||
$a,b,c$ mit $w(a,b)=1$, $w(b,c)=1$, $w(a,c)=10$; hier ist $w(a,c)=10 > 1+1 =
|
||||
w(a,b)+w(b,c)$. Ohne Dreiecksungleichung liefert das Abk\"urzen keine
|
||||
Absch\"atzung der Tourl\"ange.
|
||||
\end{itemize}
|
||||
(Eine einzige Eingabe, die \emph{beide} Eigenschaften verletzt, w\"are ebenfalls
|
||||
zul\"assig.) \textbf{G\"ute:} $\tfrac32$. \hfill$\square$
|
||||
|
||||
% ============================= A2 =============================
|
||||
\newpage
|
||||
\aufgabe{2 \quad VORLESUNGSBEWEIS -- DEFINITION NP}{2 + 8 Punkte}
|
||||
Zu zeigen: Existiert ein polynomieller Verifizierer f\"ur $L$, so existiert eine
|
||||
NDTM, die $L$ in nicht-deterministischer Polynomialzeit entscheidet.
|
||||
|
||||
\lsg
|
||||
\textbf{a) Analoge NDTM zum gegebenen \problem{VertexCover}-Verifizierer.}
|
||||
Der Verifizierer r\"at nichts, sondern \emph{pr\"uft} ein von au{\ss}en
|
||||
gegebenes Zertifikat $C$. Die analoge NDTM ersetzt dieses Zertifikat durch eine
|
||||
\emph{nichtdeterministische Ratephase} und f\"uhrt danach dieselben Pr\"ufungen
|
||||
aus:
|
||||
\begin{shaded}
|
||||
\ttfamily\small
|
||||
$M(G = (V,E),\ k)$\\
|
||||
1.\quad Rate nichtdeterministisch eine Menge von Knoten $C \subseteq V$.\\
|
||||
2.\quad Falls $|C| > k$, verwerfe.\\
|
||||
3.\quad F\"ur alle $\{u,v\} \in E$: Falls $u \notin C$ und $v \notin C$, verwerfe.\\
|
||||
4.\quad Akzeptiere.
|
||||
\end{shaded}
|
||||
\textbf{Korrektheit:} $M$ hat genau dann einen akzeptierenden Rechenweg, wenn ein
|
||||
$C \subseteq V$ existiert, das die Pr\"ufungen 2--3 \"ubersteht, d.h.\ ein
|
||||
Vertex Cover der Gr\"o{\ss}e $\le k$ -- genau dann akzeptiert auch $V$ ein
|
||||
Zertifikat. Also akzeptiert $M$ dieselbe Sprache $L$.\\
|
||||
\textbf{Laufzeit:} Raten von $C$ in $O(|V|)$, Gr\"o{\ss}e pr\"ufen $O(|V|)$,
|
||||
Kanten pr\"ufen $O(|E|)$ -- jeder Rechenweg l\"auft in $O(|V| + |E|)$, also
|
||||
polynomiell.
|
||||
|
||||
\medskip
|
||||
\textbf{b) Allgemeing\"ultiger Beweis.} Sei $L$ eine Sprache mit polynomiellem
|
||||
Verifizierer $M$ der Laufzeit $p(|x|) = O(|x|^d)$: F\"ur $x \in L$ existiert ein
|
||||
Zertifikat $c$ mit $M(x,c) = 1$; f\"ur $x \notin L$ akzeptiert $M$ kein
|
||||
Zertifikat. In $p(|x|)$ Schritten liest $M$ h\"ochstens die ersten $p(|x|)$
|
||||
Zeichen des Zertifikats.
|
||||
|
||||
\textbf{Konstruktion der NDTM $N$.} Bei Eingabe $x$ arbeitet $N$ in zwei Phasen:
|
||||
\begin{itemize}
|
||||
\item \emph{Raten:} Schreibe mit nichtdeterministischen Entscheidungen einen
|
||||
String $u$ mit $|u| \le p(|x|)$ auf ein Arbeitsband.
|
||||
\item \emph{Verifizieren:} Simuliere $M$ auf $(x,u)$ und \"ubernimm dessen
|
||||
Antwort (akzeptiere/verwirf).
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Korrektheit:}
|
||||
\begin{itemize}
|
||||
\item Sei $x \in L$. Dann existiert ein Zertifikat $c$ mit $M(x,c)=1$; davon
|
||||
liest $M$ h\"ochstens die ersten $p(|x|)$ Zeichen. Der Rechenweg, der genau
|
||||
diesen Anfang als $u$ r\"at, akzeptiert -- also akzeptiert $N$.
|
||||
\item Sei $x \notin L$. Dann verwirft $M$ jedes Zertifikat, insbesondere jedes
|
||||
geratene $u$. Also verwerfen alle Rechenwege von $N$, und $N$ verwirft.
|
||||
\end{itemize}
|
||||
Somit akzeptiert $N$ genau die Sprache $L$.
|
||||
|
||||
\textbf{Laufzeit:} Phase~1 braucht $\le p(|x|)$ Rateschritte, Phase~2 l\"auft in
|
||||
$O(|x|^d)$ -- insgesamt $O(|x|^d)$, also polynomiell. Damit entscheidet die NDTM
|
||||
$N$ die Sprache $L$ in nicht-deterministischer Polynomialzeit
|
||||
$T = O(|x|^d)$. \hfill$\square$
|
||||
|
||||
% ============================= A3 =============================
|
||||
\newpage
|
||||
\aufgabe{3 \quad NP-VOLLST\"ANDIGKEIT}{10 Punkte}
|
||||
Zu zeigen: $k$-\problem{CliqueMember} ist NP-vollst\"andig.
|
||||
|
||||
\lsg
|
||||
\textbf{$\in$ NP:} Eingabe: Graph $G = (V,E)$, Knoten $v$, Zahl $k$ und
|
||||
Zertifikat $C \subseteq V$. Der Verifizierer arbeitet wie folgt:
|
||||
\begin{itemize}
|
||||
\item Falls $|C| < k$ oder $v \notin C$, lehne ab.
|
||||
\item Pr\"ufe f\"ur jedes Paar $\{x,y\} \subseteq C$, ob $\{x,y\} \in E$; lehne
|
||||
sonst ab.
|
||||
\item Sonst akzeptiere.
|
||||
\end{itemize}
|
||||
Existiert eine Clique $C$ mit $|C| \ge k$ und $v \in C$, so ist sie ein
|
||||
akzeptiertes Zertifikat; sonst scheitert jedes Zertifikat an einer Pr\"ufung.
|
||||
Laufzeit: Gr\"o{\ss}e und Mitgliedschaft pr\"ufen $O(|V|)$, alle Paare testen
|
||||
$O(|V|^2)$ -- gesamt $O(|V|^2)$. Also gilt $k$-\problem{CliqueMember} $\in \NP$.
|
||||
|
||||
\textbf{NP-schwer:} Zeige die NP-Schwere durch die Reduktion
|
||||
\[ \problem{Clique} \redp k\text{-}\problem{CliqueMember}. \]
|
||||
\problem{Clique} ist bereits als NP-vollst\"andig bekannt.
|
||||
|
||||
\textbf{Konstruktion:} Sei $(G = (V,E),\, k)$ eine \problem{Clique}-Instanz. Wir
|
||||
konstruieren daraus eine $k$-\problem{CliqueMember}-Instanz
|
||||
$(G' = (V', E'),\, v,\, k')$ durch:
|
||||
\begin{itemize}
|
||||
\item $V' = V \cup \{v\}$ mit einem neuen Knoten $v \notin V$
|
||||
\item $E' = E \cup \{\{v,w\} \mid w \in V\}$ ($v$ ist mit allen Knoten aus $V$
|
||||
verbunden, also universell)
|
||||
\item $k' = k + 1$, ausgezeichneter Knoten $v$
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Beweis hin:}
|
||||
\begin{itemize}
|
||||
\item Sei $C$ eine Clique in $G$ mit $|C| \ge k$.
|
||||
\item Da $v$ mit allen Knoten von $V \supseteq C$ verbunden ist, sind alle Paare
|
||||
in $C \cup \{v\}$ Kanten in $E'$; also ist $C \cup \{v\}$ eine Clique in $G'$.
|
||||
\item Wegen $v \notin V \supseteq C$ gilt $|C \cup \{v\}| = |C| + 1 \ge k + 1 =
|
||||
k'$, und $v \in C \cup \{v\}$.
|
||||
\item Also ist $(G', v, k')$ eine Ja-Instanz.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Beweis zur\"uck:}
|
||||
\begin{itemize}
|
||||
\item Sei $C'$ eine Clique in $G'$ mit $|C'| \ge k'$ und $v \in C'$.
|
||||
\item Setze $C = C' \setminus \{v\}$; wegen $v \in C'$ gilt $|C| = |C'| - 1 \ge
|
||||
k' - 1 = k$, und $C \subseteq V$.
|
||||
\item Alle Kanten zwischen Knoten von $C$ liegen in $E$, denn neu in $E'$ sind
|
||||
nur die Kanten mit Endpunkt $v$, und $v \notin C$.
|
||||
\item Also ist $C$ eine Clique in $G$ mit $|C| \ge k$.
|
||||
\end{itemize}
|
||||
(Beide Richtungen zusammen: $(G',v,k')$ ist Ja-Instanz $\iff$ $(G,k)$ ist
|
||||
Ja-Instanz. Der Randfall $k \ge 1$ ist gew\"ahrleistet, da $k' = k+1 \ge 2$ und
|
||||
$v$ stets in der Clique liegen \emph{muss}.)
|
||||
|
||||
\textbf{Laufzeit:} Einen Knoten und $|V|$ Kanten anlegen -- $O(|V| + |E|)$,
|
||||
polynomiell.
|
||||
|
||||
Da $k$-\problem{CliqueMember} $\in \NP$ und NP-schwer ist, folgt:
|
||||
$k$-\problem{CliqueMember} ist NP-vollst\"andig. \hfill$\square$
|
||||
|
||||
% ============================= A4 =============================
|
||||
\newpage
|
||||
\aufgabe{4 \quad ETH}{1 + 4 + 1 + 4 Punkte}
|
||||
|
||||
\lsg
|
||||
\textbf{a) Parameter der \problem{VertexCover}-Instanz.} Es gilt $V' = V$, also
|
||||
\[ |V'| = |V|, \qquad
|
||||
|E'| = \binom{|V|}{2} - |E| = \tfrac12\,|V|\,(|V|-1) - |E| \;\le\; |V|^2. \]
|
||||
(Die Komplementkantenmenge $E'$ enth\"alt alle $\binom{|V|}{2}$ m\"oglichen
|
||||
Kanten au{\ss}er den $|E|$ Kanten von $G$.)
|
||||
|
||||
\medskip
|
||||
\textbf{b) ETH-Schranken f\"ur \problem{VertexCover}.} Die Reduktion
|
||||
$\problem{Clique} \redp \problem{VertexCover}$ l\"auft in Polynomialzeit.
|
||||
|
||||
\textbf{Bez\"uglich $|V'|$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, \problem{VertexCover} w\"are in $2^{o(|V'|)} \cdot |I|^{O(1)}$
|
||||
l\"osbar.
|
||||
\item Da die Reduktion polynomiell ist und $|V'| = |V|$ gilt, erg\"abe die
|
||||
Komposition (Reduktion, dann Algorithmus) einen $2^{o(|V|)} \cdot |I|^{O(1)}$-Algorithmus
|
||||
f\"ur \problem{Clique}.
|
||||
\item Widerspruch zur ETH (\problem{Clique} $\notin 2^{o(|V|)} \cdot |I|^{O(1)}$).
|
||||
\item Also besitzt \problem{VertexCover} keinen $2^{o(|V'|)} \cdot |I|^{O(1)}$-Algorithmus.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Bez\"uglich $|E'|$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, \problem{VertexCover} w\"are in $2^{o(\sqrt{|E'|})} \cdot |I|^{O(1)}$
|
||||
l\"osbar.
|
||||
\item Wegen $|E'| \le \tfrac12|V|(|V|-1) \le |V|^2$ ist $\sqrt{|E'|} \le |V|$,
|
||||
also $o(\sqrt{|E'|}) \subseteq o(|V|)$.
|
||||
\item Damit w\"are \problem{VertexCover} in $2^{o(|V|)} \cdot |I|^{O(1)}$
|
||||
l\"osbar, und mit $|V'|=|V|$ (Reduktion polynomiell) folgt ein
|
||||
$2^{o(|V|)} \cdot |I|^{O(1)}$-Algorithmus f\"ur \problem{Clique}.
|
||||
\item Widerspruch zur ETH.
|
||||
\item Also besitzt \problem{VertexCover} keinen
|
||||
$2^{o(\sqrt{|E'|})} \cdot |I|^{O(1)}$-Algorithmus.
|
||||
\end{itemize}
|
||||
Ergebnis: kein $2^{o(|V'|)}$ und kein $2^{o(\sqrt{|E'|})}$.
|
||||
|
||||
\medskip
|
||||
\textbf{c) Parameter der \problem{HittingSet}-Instanz.} Die Grundmenge besteht
|
||||
aus den $2n$ Literalen, und es gibt $n$ Paar-Mengen sowie $m$ Klausel-Mengen:
|
||||
\[ |U| = 2n, \qquad r = n + m. \]
|
||||
|
||||
\medskip
|
||||
\textbf{d) ETH-Schranken f\"ur \problem{HittingSet}.} Die Reduktion
|
||||
$\problem{3-SAT} \redp \problem{HittingSet}$ l\"auft in Polynomialzeit. (Sie ist
|
||||
korrekt, weil die $n$ disjunkten Paar-Mengen bei $k=n$ erzwingen, dass $H$ aus
|
||||
jedem Paar $\{x_i,\bar x_i\}$ \emph{genau} ein Literal enth\"alt -- also eine
|
||||
Belegung kodiert --, und das Treffen der Klausel-Mengen genau der Erf\"ullung der
|
||||
Klauseln entspricht.)
|
||||
|
||||
\textbf{Bez\"uglich $|U|$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, \problem{HittingSet} w\"are in $2^{o(|U|)} \cdot |I|^{O(1)}$
|
||||
l\"osbar.
|
||||
\item Wegen $|U| = 2n$ ist $o(|U|) = o(n)$; die Komposition erg\"abe einen
|
||||
$2^{o(n)} \cdot |I|^{O(1)}$-Algorithmus f\"ur \problem{3-SAT}.
|
||||
\item Direkter Widerspruch zur ETH.
|
||||
\item Also kein $2^{o(|U|)} \cdot |I|^{O(1)}$-Algorithmus.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Bez\"uglich $r$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, \problem{HittingSet} w\"are in $2^{o(r)} \cdot |I|^{O(1)}$
|
||||
l\"osbar.
|
||||
\item Wegen $r = n + m$ und $n \le 3m$ gilt $r \le 4m = O(m)$, also
|
||||
$o(r) \subseteq o(m)$; die Komposition erg\"abe einen
|
||||
$2^{o(m)} \cdot |I|^{O(1)}$-Algorithmus f\"ur \problem{3-SAT}.
|
||||
\item Widerspruch zur ETH nach dem Sparsification-Lemma (\problem{3-SAT}
|
||||
$\notin 2^{o(m)} \cdot |I|^{O(1)}$).
|
||||
\item Also kein $2^{o(r)} \cdot |I|^{O(1)}$-Algorithmus.
|
||||
\end{itemize}
|
||||
Ergebnis: kein $2^{o(|U|)}$ und kein $2^{o(r)}$. \hfill$\square$
|
||||
|
||||
% ============================= A5 =============================
|
||||
\newpage
|
||||
\aufgabe{5 \quad APPROXIMATIVE ALGORITHMEN -- MAX-3-SAT}{2 + 5 + 3 + (10) Punkte}
|
||||
|
||||
\lsg
|
||||
\textbf{a) Anwendung auf $\varphi$.} Dabei setzt $\beta_0$ alle Variablen auf
|
||||
\false\ und $\beta_1$ alle auf \true. Klauselauswertung:
|
||||
\begin{center}
|
||||
\begin{tabular}{l|c|c}
|
||||
Klausel & unter $\beta_0$ & unter $\beta_1$\\\hline
|
||||
$C_1 = (x_1 \vee x_2 \vee x_3)$ & \false\ (alle Lit.\ falsch) & \true\\
|
||||
$C_2 = (\neg x_1 \vee \neg x_2 \vee \neg x_3)$ & \true\ ($\neg x_1$) & \false\\
|
||||
$C_3 = (x_1 \vee \neg x_2 \vee x_3)$ & \true\ ($\neg x_2$) & \true\\
|
||||
$C_4 = (\neg x_1 \vee x_2 \vee \neg x_3)$ & \true\ ($\neg x_1$) & \true\\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Also $v(\beta_0) = 3$ und $v(\beta_1) = 3$. Der Algorithmus gibt (bei
|
||||
Gleichstand $\beta_0$) den Wert $v(A(\varphi)) = \max\{3,3\} = 3$ aus.
|
||||
|
||||
\emph{Optimum (Brute-Force \"uber alle $8$ Belegungen):} Die Belegung
|
||||
$(x_1,x_2,x_3) = (\false, \false, \true)$ erf\"ullt $C_1$ (durch $x_3$), $C_2$
|
||||
(durch $\neg x_1$), $C_3$ (durch $\neg x_2$) und $C_4$ (durch $\neg x_1$), also
|
||||
alle vier. Somit $v(\mathrm{OPT}(\varphi)) = 4$. (Verh\"altnis
|
||||
$\tfrac{\mathrm{OPT}}{A} = \tfrac43$.)
|
||||
|
||||
\medskip
|
||||
\textbf{b) G\"ute $2$.} Zu zeigen: $v(A(\varphi)) \ge \tfrac12\,
|
||||
v(\mathrm{OPT}(\varphi))$. Sei $\varphi$ eine Formel mit $m$ Klauseln.
|
||||
|
||||
\textbf{Jede Klausel z\"ahlt mindestens einmal.} Unter $\beta_0$ ist jedes
|
||||
\emph{negative} Literal wahr, unter $\beta_1$ jedes \emph{positive}. Da jede
|
||||
Klausel mindestens ein Literal enth\"alt und jedes Literal positiv oder negativ
|
||||
ist, wird jede Klausel von $\beta_0$ oder von $\beta_1$ (oder beiden) erf\"ullt.
|
||||
Somit z\"ahlt jede Klausel in der Summe mindestens einmal:
|
||||
\[ v(\beta_0) + v(\beta_1) \;\ge\; m. \]
|
||||
|
||||
\textbf{Obere Schranke f\"ur das Optimum.} H\"ochstens alle $m$ Klauseln k\"onnen
|
||||
erf\"ullt sein, also $v(\mathrm{OPT}(\varphi)) \le m$.
|
||||
|
||||
\textbf{Kombination.} $A$ w\"ahlt die bessere der beiden Belegungen, daher
|
||||
\[ v(A(\varphi)) = \max\{v(\beta_0), v(\beta_1)\}
|
||||
\ge \frac{v(\beta_0) + v(\beta_1)}{2}
|
||||
\ge \frac{m}{2}
|
||||
\ge \frac{v(\mathrm{OPT}(\varphi))}{2}. \]
|
||||
Also hat $A$ die G\"ute $2$.
|
||||
|
||||
\medskip
|
||||
\textbf{c) Scharfe Instanz.} W\"ahle
|
||||
\[ \varphi^\ast = (x_1 \vee x_2 \vee x_3) \wedge (\neg x_1 \vee \neg x_2 \vee \neg x_3). \]
|
||||
\begin{itemize}
|
||||
\item Unter $\beta_0$ (alles \false) ist die erste Klausel unerf\"ullt (alle
|
||||
Literale positiv, also falsch) und die zweite erf\"ullt: $v(\beta_0) = 1$.
|
||||
\item Unter $\beta_1$ (alles \true) ist die zweite Klausel unerf\"ullt und die
|
||||
erste erf\"ullt: $v(\beta_1) = 1$.
|
||||
\item Also $v(A(\varphi^\ast)) = \max\{1,1\} = 1$.
|
||||
\item Die Belegung $(x_1,x_2,x_3) = (\true, \false, \false)$ erf\"ullt beide
|
||||
Klauseln ($x_1$ bzw.\ $\neg x_2$), also $v(\mathrm{OPT}(\varphi^\ast)) = 2$.
|
||||
\end{itemize}
|
||||
Damit gilt $v(A(\varphi^\ast)) = 1 = \tfrac12 \cdot 2 = \tfrac12\,
|
||||
v(\mathrm{OPT}(\varphi^\ast))$: die Schranke aus b) wird mit Gleichheit
|
||||
angenommen, die G\"ute $2$ ist scharf.
|
||||
|
||||
\medskip
|
||||
\textbf{d) BONUS -- \problem{MAX-$k$-SAT}.} \emph{Behauptung:} F\"ur jedes
|
||||
$k \ge 2$ garantiert derselbe Algorithmus $A$ ebenfalls die G\"ute $2$, und diese
|
||||
ist scharf.
|
||||
|
||||
\textbf{G\"ute $2$ (Beweis).} Das Argument aus b) benutzt nur, dass jede Klausel
|
||||
mindestens ein Literal enth\"alt und jedes Literal positiv oder negativ ist -- es
|
||||
ist von der Klausell\"ange unabh\"angig. Auch bei genau $k$ Literalen pro Klausel
|
||||
gilt: eine Klausel wird von $\beta_0$ nur dann nicht erf\"ullt, wenn all ihre
|
||||
$k$ Literale positiv sind; von $\beta_1$ nur dann nicht, wenn alle $k$ negativ
|
||||
sind. Beides zugleich ist bei $k \ge 1$ unm\"oglich. Also wird jede Klausel von
|
||||
$\beta_0$ oder $\beta_1$ erf\"ullt, mithin $v(\beta_0) + v(\beta_1) \ge m$, und
|
||||
wie in b)
|
||||
\[ v(A(\varphi)) = \max\{v(\beta_0), v(\beta_1)\} \ge \frac{m}{2}
|
||||
\ge \frac{v(\mathrm{OPT}(\varphi))}{2}. \]
|
||||
Die garantierte G\"ute ist also $2$, unabh\"angig von $k$.
|
||||
|
||||
\textbf{Sch\"arfe (f\"ur jedes $k$).} Betrachte \"uber den Variablen
|
||||
$x_1,\dots,x_k$ die beiden Klauseln
|
||||
\[ \varphi_k^\ast = (x_1 \vee \dots \vee x_k) \wedge
|
||||
(\neg x_1 \vee \dots \vee \neg x_k). \]
|
||||
Jede Klausel besteht aus genau $k$ Literalen mit paarweise verschiedenen
|
||||
Variablen. $\beta_0$ erf\"ullt nur die zweite, $\beta_1$ nur die erste, also
|
||||
$v(A(\varphi_k^\ast)) = 1$. Die Belegung $x_1 = \true$, $x_2 = \dots = x_k =
|
||||
\false$ erf\"ullt beide Klauseln, also $v(\mathrm{OPT}(\varphi_k^\ast)) = 2$.
|
||||
Damit $v(A(\varphi_k^\ast)) = \tfrac12\,v(\mathrm{OPT}(\varphi_k^\ast))$, die
|
||||
G\"ute $2$ wird f\"ur jedes $k$ mit Gleichheit angenommen und ist somit
|
||||
scharf. \hfill$\square$
|
||||
|
||||
\end{document}
|
||||
2
lernmaterial/loesung_probeklausur2.aux
Normal file
2
lernmaterial/loesung_probeklausur2.aux
Normal file
@@ -0,0 +1,2 @@
|
||||
\relax
|
||||
\gdef \@abspage@last{7}
|
||||
171
lernmaterial/loesung_probeklausur2.fdb_latexmk
Normal file
171
lernmaterial/loesung_probeklausur2.fdb_latexmk
Normal file
@@ -0,0 +1,171 @@
|
||||
# Fdb version 4
|
||||
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L/lmm/m/it/10.95 k$ \T1/lmr/m/n/10.95 paar-weise dis-junk-ten Kan-ten $\OMS/lms
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[]
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[7] (./loesung_probeklausur2.aux)
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***********
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LaTeX2e <2023-11-01> patch level 1
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lernmaterial/loesung_probeklausur2.pdf
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lernmaterial/loesung_probeklausur2.tex
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\documentclass[11pt,a4paper]{article}
|
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\usepackage[T1]{fontenc}
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\usepackage{lmodern}
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\usepackage[utf8]{inputenc}
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\usepackage[margin=2.5cm]{geometry}
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\usepackage{amsmath,amssymb}
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\usepackage{xcolor}
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\usepackage{tikz}
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\emergencystretch=1.5em
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\definecolor{shadecolor}{gray}{0.94}
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% ---- graue Box fuer Pseudocode/Problem-Boxen (nur mit xcolor) ----
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\makeatletter
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\newsavebox{\shadedbox}
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\newenvironment{shaded}%
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{\par\smallskip\noindent
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\colorbox{shadecolor}{\usebox{\shadedbox}}\par\smallskip}
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\makeatother
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% ---- alphabetische Aufzaehlung a) b) c) ohne enumitem ----
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\renewcommand{\theenumi}{\alph{enumi}}
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\renewcommand{\labelenumi}{\theenumi)}
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|
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% ---- Makros ----
|
||||
\newcommand{\problem}[1]{\textsc{#1}}
|
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\newcommand{\redp}{\le_p}
|
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\newcommand{\NP}{\textsf{NP}}
|
||||
\newcommand{\true}{\textsf{wahr}}
|
||||
\newcommand{\false}{\textsf{falsch}}
|
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\tikzset{knoten/.style={circle,draw,thick,minimum size=6.5mm,inner sep=1pt}}
|
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|
||||
% ---- Aufgabenkopf ----
|
||||
\newcommand{\aufgabe}[2]{\clearpage
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||||
\noindent{\large\textbf{Aufgabe #1}}\hfill\textbf{(#2)}\par\nobreak
|
||||
\noindent\rule{\linewidth}{0.4pt}\par\medskip}
|
||||
|
||||
\begin{document}
|
||||
|
||||
% ================= Kopf =================
|
||||
\begin{center}
|
||||
{\large\textbf{Christian-Albrechts-Universität zu Kiel}}\\[2pt]
|
||||
Institut für Informatik, Arbeitsgruppe Algorithmen und Komplexität\\
|
||||
Prof.\ Dr.\ K.\ Jansen\\[10pt]
|
||||
{\large\textbf{Musterlösung -- Probeklausur 2}}\\[3pt]
|
||||
{\large\textbf{Analyse von Algorithmen und Komplexität}}\\[2pt]
|
||||
SS 2026
|
||||
\end{center}
|
||||
|
||||
% ======================================================================
|
||||
\aufgabe{1 \quad ANWENDUNG -- LPT}{5 + 2 + 3 Punkte}
|
||||
|
||||
\textbf{a)} \problem{LPT} sortiert die Jobs absteigend und wendet darauf
|
||||
\problem{ListScheduling} an; Lastvektor $(M_1, M_2, M_3)$, bei Gleichstand
|
||||
kleinster Index.
|
||||
\begin{itemize}
|
||||
\item Absteigend sortiert: $5,\ 5,\ 4,\ 4,\ 3,\ 3,\ 3$.
|
||||
\item $5 \to M_1$: $(5,0,0)$; \quad $5 \to M_2$: $(5,5,0)$; \quad
|
||||
$4 \to M_3$: $(5,5,4)$.
|
||||
\item $4 \to M_3$ (kleinste Last $4$): $(5,5,8)$.
|
||||
\item $3 \to M_1$ (kleinste Last $5$, Gleichstand $M_1,M_2$): $(8,5,8)$.
|
||||
\item $3 \to M_2$ (kleinste Last $5$): $(8,8,8)$.
|
||||
\item $3 \to M_1$ (kleinste Last $8$, Gleichstand, kleinster Index): $(11,8,8)$.
|
||||
\end{itemize}
|
||||
Belegung: $M_1 = \{5,3,3\}$ (Last $11$), $M_2 = \{5,3\}$ (Last $8$),
|
||||
$M_3 = \{4,4\}$ (Last $8$). Makespan $\mathrm{LPT} = C_{\max} = 11$.
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[x=0.72cm,y=0.9cm]
|
||||
\draw[->] (0,-0.35) -- (12,-0.35) node[right]{$t$};
|
||||
\foreach \t in {0,2,4,6,8,10} \draw (\t,-0.3)--(\t,-0.4) node[below,font=\scriptsize]{$\t$};
|
||||
\draw (11,-0.3)--(11,-0.4) node[below,font=\scriptsize]{$11$};
|
||||
\node[left] at (0,2.4) {$M_1$};
|
||||
\draw[fill=black!12] (0,2) rectangle (5,2.8); \node at (2.5,2.4){$5$};
|
||||
\draw[fill=black!20] (5,2) rectangle (8,2.8); \node at (6.5,2.4){$3$};
|
||||
\draw[fill=black!12] (8,2) rectangle (11,2.8); \node at (9.5,2.4){$3$};
|
||||
\node[left] at (0,1.4) {$M_2$};
|
||||
\draw[fill=black!12] (0,1) rectangle (5,1.8); \node at (2.5,1.4){$5$};
|
||||
\draw[fill=black!20] (5,1) rectangle (8,1.8); \node at (6.5,1.4){$3$};
|
||||
\node[left] at (0,0.4) {$M_3$};
|
||||
\draw[fill=black!12] (0,0) rectangle (4,0.8); \node at (2,0.4){$4$};
|
||||
\draw[fill=black!20] (4,0) rectangle (8,0.8); \node at (6,0.4){$4$};
|
||||
\draw[dashed] (11,-0.35)--(11,2.9);
|
||||
\node[right,font=\small] at (11.1,1.4){$C_{\max}=11$};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\emph{Bemerkung:} Wegen $\sum_j p_j = 27 = 3 \cdot 9$ und der Aufteilung
|
||||
$\{5,4\},\{5,4\},\{3,3,3\}$ (Lasten $9,9,9$) ist $\mathrm{OPT} = 9$. \problem{LPT}
|
||||
ist hier also \emph{nicht} optimal: die Rate ist $\tfrac{11}{9}$.
|
||||
|
||||
\textbf{b)} \problem{LPT} hat die approximative Güte
|
||||
\[
|
||||
\frac{4}{3} - \frac{1}{3m}.
|
||||
\]
|
||||
Für $m = 3$ ergibt das $\frac43 - \frac19 = \frac{11}{9}$; die Instanz aus a)
|
||||
erreicht diese Schranke also exakt und ist ein worst-case-Beispiel.
|
||||
|
||||
\textbf{c)} Instanz mit $m = 2$ und Jobs in der Reihenfolge $p = (1, 1, 2)$.
|
||||
\problem{ListScheduling} legt jeden Job auf die aktuell leerste Maschine
|
||||
(Lastvektor $(M_1, M_2)$):
|
||||
\begin{itemize}
|
||||
\item $1 \to M_1$: $(1,0)$;\quad $1 \to M_2$: $(1,1)$;\quad
|
||||
$2 \to M_1$ (Gleichstand, kleinster Index): $(3,1)$.
|
||||
\end{itemize}
|
||||
$\mathrm{LS} = 3$. Optimal ist $\{1,1\}$ auf einer Maschine und $\{2\}$ auf der
|
||||
anderen (Lasten $2,2$), also $\mathrm{OPT} = 2$. Die Rate ist
|
||||
$\frac{\mathrm{LS}}{\mathrm{OPT}} = \frac{3}{2} = 2 - \frac12 = 2 - \frac1m$.
|
||||
\hfill$\square$
|
||||
|
||||
% ======================================================================
|
||||
\aufgabe{2 \quad VORLESUNGSBEWEIS -- TRANSITIVITÄT}{10 Punkte}
|
||||
|
||||
Zu zeigen: Aus $L_1 \redp L_2$ und $L_2 \redp L_3$ folgt $L_1 \redp L_3$.
|
||||
|
||||
\textbf{Voraussetzung.} Sei $R_1$ eine Polynomialzeitreduktion $L_1 \redp L_2$
|
||||
mit Laufzeit $O(n^a)$ und $R_2$ eine Polynomialzeitreduktion $L_2 \redp L_3$
|
||||
mit Laufzeit $O(n^b)$, wobei $a, b \ge 1$ konstant sind. Nach Definition gilt
|
||||
für alle $x, y$:
|
||||
\[
|
||||
x \in L_1 \iff R_1(x) \in L_2, \qquad y \in L_2 \iff R_2(y) \in L_3.
|
||||
\]
|
||||
|
||||
\textbf{Konstruktion.} Definiere die Komposition $R := R_2 \circ R_1$, also
|
||||
$R(x) = R_2(R_1(x))$: Berechne zuerst $R_1(x)$, wende darauf $R_2$ an.
|
||||
|
||||
\textbf{Korrektheit.} Für jede Eingabe $x$ gilt
|
||||
\[
|
||||
x \in L_1 \;\iff\; R_1(x) \in L_2 \;\iff\; R_2(R_1(x)) \in L_3,
|
||||
\]
|
||||
wobei die erste Äquivalenz aus der Korrektheit von $R_1$ und die zweite aus der
|
||||
Korrektheit von $R_2$ (angewandt auf $y = R_1(x)$) folgt. Insgesamt also
|
||||
$x \in L_1 \iff R(x) \in L_3$ -- in beiden Richtungen.
|
||||
|
||||
\textbf{Laufzeit.} Sei $|x| = n$.
|
||||
\begin{itemize}
|
||||
\item Die Berechnung von $R_1(x)$ braucht $O(n^a)$ Zeit. Da in dieser Zeit
|
||||
höchstens $O(n^a)$ Symbole geschrieben werden, hat die Ausgabe $R_1(x)$ die
|
||||
Größe $|R_1(x)| = O(n^a)$.
|
||||
\item Auf dieser Eingabe der Größe $O(n^a)$ braucht $R_2$ die Zeit
|
||||
$O\bigl((n^a)^b\bigr) = O(n^{ab})$.
|
||||
\item Gesamtlaufzeit $O(n^a) + O(n^{ab}) = O(n^{ab})$; wegen $a, b \ge 1$ ist
|
||||
$n^{ab}$ ein Polynom in $n$.
|
||||
\end{itemize}
|
||||
Damit ist $R$ eine in Polynomialzeit berechenbare Funktion mit
|
||||
$x \in L_1 \iff R(x) \in L_3$, also eine Reduktion $L_1 \redp L_3$. \hfill$\square$
|
||||
|
||||
% ======================================================================
|
||||
\aufgabe{3 \quad NP-VOLLSTÄNDIGKEIT}{10 Punkte}
|
||||
Sei $L$ das Problem \problem{SubsetSum-ohne-Dreierpotenzen}.
|
||||
|
||||
\textbf{$\in$ NP:} Eingabe: Zahlen $c_1, \dots, c_n$, Zielwert $K$ und
|
||||
Zertifikat $S \subseteq [n]$. Der Verifizierer arbeitet wie folgt:
|
||||
\begin{itemize}
|
||||
\item Prüfe für jede Größe $c_i$, dass sie keine Dreierpotenz ist (teile $c_i$
|
||||
so lange wie möglich durch $3$; ist das Ergebnis $\ne 1$ bzw.\ tritt ein Rest
|
||||
auf, so ist $c_i$ keine Dreierpotenz); lehne sonst ab.
|
||||
\item Prüfe $\sum_{i \in S} c_i = K$; lehne sonst ab.
|
||||
\item Sonst akzeptiere.
|
||||
\end{itemize}
|
||||
\textbf{Korrektheit:} Ist die Instanz eine gültige Variante mit Lösung, so ist
|
||||
diese ein akzeptiertes Zertifikat; andernfalls scheitert jedes Zertifikat an
|
||||
einer Prüfung.
|
||||
|
||||
\textbf{Laufzeit:} Je Größe der Dreier-Test $O(\log c_i)$ und die Summe bilden
|
||||
$O(n)$ -- polynomiell. Also gilt $L \in \NP$.
|
||||
|
||||
\textbf{NP-schwer:} Zeige die NP-Schwere durch die Reduktion
|
||||
\[
|
||||
\problem{SubsetSum} \redp L.
|
||||
\]
|
||||
\problem{SubsetSum} ist bereits als NP-vollständig bekannt.
|
||||
|
||||
\textbf{Konstruktion:} Sei $(c_1, \dots, c_n, K)$ eine
|
||||
\problem{SubsetSum}-Instanz mit $c_i \in \mathbb{N}_{>0}$. Skaliere mit $2$:
|
||||
\[
|
||||
c_i' := 2 c_i \ \ (i \in [n]), \qquad K' := 2K.
|
||||
\]
|
||||
Jede neue Größe $c_i' = 2 c_i \ge 2$ ist gerade. Dreierpotenzen $3^t$ sind
|
||||
dagegen für jedes $t \in \mathbb{N}_0$ ungerade (Produkt ungerader Faktoren
|
||||
bzw.\ $3^0 = 1$). Eine gerade Zahl $\ge 2$ ist also nie eine Dreierpotenz;
|
||||
insbesondere ist $c_i' \ne 1$ und $c_i' \ne 3^t$ für alle $t \ge 1$. Damit
|
||||
liegt die konstruierte Instanz $(c_1', \dots, c_n', K')$ in $L$.
|
||||
|
||||
\textbf{Beweis SubsetSum nach Variante:}
|
||||
\begin{itemize}
|
||||
\item Sei $S$ mit $\sum_{i \in S} c_i = K$.
|
||||
\item Dann gilt $\sum_{i \in S} c_i' = \sum_{i \in S} 2 c_i
|
||||
= 2 \sum_{i \in S} c_i = 2K = K'$.
|
||||
\item Also ist $S$ eine Lösung der Variante.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Beweis Variante nach SubsetSum:}
|
||||
\begin{itemize}
|
||||
\item Sei $S$ mit $\sum_{i \in S} c_i' = K'$.
|
||||
\item Das heißt $2 \sum_{i \in S} c_i = 2K$.
|
||||
\item Division durch $2$ liefert $\sum_{i \in S} c_i = K$.
|
||||
\item Also ist $S$ eine \problem{SubsetSum}-Lösung.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Laufzeit:} Jede der $n$ Zahlen und $K$ mit $2$ multiplizieren -- $O(n)$.
|
||||
|
||||
Da \problem{SubsetSum} NP-vollständig ist, $\problem{SubsetSum} \redp L$ gilt
|
||||
und $L \in \NP$, ist $L = \problem{SubsetSum-ohne-Dreierpotenzen}$
|
||||
NP-vollständig. \hfill$\square$
|
||||
|
||||
% ======================================================================
|
||||
\aufgabe{4 \quad ETH}{1 + 4 + 1 + 4 Punkte}
|
||||
|
||||
\textbf{Teil 1: $k$-\problem{IndependentSet}.}
|
||||
|
||||
\textbf{a)} Der Komplementgraph hat dieselben Knoten und als Kanten genau die
|
||||
Nicht-Kanten von $G$:
|
||||
\[
|
||||
|V'| = |V|, \qquad
|
||||
|E'| = \binom{|V|}{2} - |E| = \tfrac12 |V|(|V|-1) - |E| \ \le\ |V|^2.
|
||||
\]
|
||||
|
||||
\textbf{b)}
|
||||
|
||||
\textbf{Bzgl.\ $|V'|$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, $k$-\problem{IndependentSet} wäre in
|
||||
$2^{o(|V'|)} \cdot |I|^{O(1)}$ lösbar.
|
||||
\item Ein Independent Set in $G' = \bar G$ ist genau eine Clique in $G$; die
|
||||
Reduktion ist in Polynomialzeit berechenbar. Der Algorithmus löst also
|
||||
$k$-\problem{Clique}.
|
||||
\item Wegen $|V'| = |V|$ wird daraus ein $2^{o(|V|)} \cdot |I|^{O(1)}$-Algorithmus
|
||||
für $k$-\problem{Clique}.
|
||||
\item Widerspruch zum Hinweis.
|
||||
\end{itemize}
|
||||
|
||||
\textbf{Bzgl.\ $|E'|$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, $k$-\problem{IndependentSet} wäre in
|
||||
$2^{o(\sqrt{|E'|})} \cdot |I|^{O(1)}$ lösbar.
|
||||
\item Wegen $|E'| \le |V|^2$ ist $\sqrt{|E'|} \le |V|$, also
|
||||
$\sqrt{|E'|} = O(|V|)$ und damit $o(\sqrt{|E'|}) \subseteq o(|V|)$.
|
||||
\item Via Komplementbildung wird daraus ein $2^{o(|V|)} \cdot |I|^{O(1)}$-Algorithmus
|
||||
für $k$-\problem{Clique}.
|
||||
\item Widerspruch zum Hinweis.
|
||||
\end{itemize}
|
||||
Bounds: kein $2^{o(|V'|)}$, kein $2^{o(\sqrt{|E'|})}$.
|
||||
|
||||
\medskip
|
||||
\textbf{Teil 2: \problem{SubsetSum}.}
|
||||
|
||||
\textbf{c)} Da die Reduktion pro Menge $S_j \in F$ genau ein Item erzeugt, gilt
|
||||
für die Itemzahl
|
||||
\[
|
||||
n = |F|.
|
||||
\]
|
||||
|
||||
\textbf{d)}
|
||||
|
||||
\textbf{Bzgl.\ $n$:}
|
||||
\begin{itemize}
|
||||
\item Angenommen, \problem{SubsetSum} wäre in
|
||||
$2^{o(\sqrt[4]{n})} \cdot |I|^{O(1)}$ lösbar.
|
||||
\item Die Reduktion $\problem{3-ExactCover} \redp \problem{SubsetSum}$ ist in
|
||||
Polynomialzeit berechenbar und erzeugt eine Instanz mit $n = |F|$ Items.
|
||||
\item Verkettet man sie mit dem angenommenen Algorithmus, so entsteht wegen
|
||||
$n = |F|$ ein $2^{o(\sqrt[4]{|F|})} \cdot |I|^{O(1)}$-Algorithmus für
|
||||
\problem{3-ExactCover}.
|
||||
\item Widerspruch zum Hinweis.
|
||||
\end{itemize}
|
||||
Bound: kein $2^{o(\sqrt[4]{n})}$. \hfill$\square$
|
||||
|
||||
% ======================================================================
|
||||
\aufgabe{5 \quad APPROXIMATIVE ALGORITHMEN -- 2ApproxVC}{2 + 5 + 3 + (10) Punkte}
|
||||
|
||||
\textbf{a)} Der Algorithmus fügt bei einer Kante beide Endpunkte hinzu, falls
|
||||
beide noch frei (nicht in $C$) sind. Kantenreihenfolge
|
||||
$\{1,2\}, \{1,3\}, \{2,3\}, \{3,4\}, \{4,5\}, \{4,6\}, \{5,6\}$:
|
||||
\begin{itemize}
|
||||
\item $\{1,2\}$: beide frei $\Rightarrow$ aufnehmen, $C = \{1,2\}$.
|
||||
\item $\{1,3\}$: $1 \in C$ $\Rightarrow$ überspringen.
|
||||
\item $\{2,3\}$: $2 \in C$ $\Rightarrow$ überspringen.
|
||||
\item $\{3,4\}$: beide frei $\Rightarrow$ aufnehmen, $C = \{1,2,3,4\}$.
|
||||
\item $\{4,5\}$: $4 \in C$ $\Rightarrow$ überspringen.
|
||||
\item $\{4,6\}$: $4 \in C$ $\Rightarrow$ überspringen.
|
||||
\item $\{5,6\}$: beide frei $\Rightarrow$ aufnehmen, $C = \{1,2,3,4,5,6\}$.
|
||||
\end{itemize}
|
||||
Der Algorithmus liefert $C = \{1,2,3,4,5,6\}$ mit $|C| = 6$.
|
||||
|
||||
\emph{Vergleich mit dem Optimum:} $G$ besteht aus den beiden knotendisjunkten
|
||||
Dreiecken $\{1,2,3\}$ und $\{4,5,6\}$ (verbunden durch die Brücke $\{3,4\}$).
|
||||
Jedes Dreieck erzwingt mindestens $2$ Knoten im Cover (bei nur einem bleibt eine
|
||||
Kante ungedeckt), und die Dreiecke sind disjunkt, also $|C^\ast| \ge 4$. Die
|
||||
Menge $\{1,2,4,5\}$ deckt alle Kanten ab (auch die Brücke via $4$), somit
|
||||
$|C^\ast| = 4$. Die Rate in diesem Fall ist $\frac{|C|}{|C^\ast|} = \frac{6}{4}
|
||||
= \frac{3}{2}$.
|
||||
|
||||
\textbf{b)} Zu zeigen: $|C| \le 2\,|C^\ast|$ für ein minimales Vertex Cover
|
||||
$C^\ast$. Sei $A$ die Menge der Kanten, bei denen \emph{beide} Endpunkte
|
||||
aufgenommen wurden; da jede solche Kante genau zwei neue Knoten beisteuert und
|
||||
kein Knoten doppelt aufgenommen wird, gilt $|C| = 2|A|$.
|
||||
|
||||
\textbf{$A$ ist ein Matching.} Hätten zwei Kanten aus $A$ einen gemeinsamen
|
||||
Endpunkt $w$, so wäre bei der später betrachteten dieser Kanten der Knoten $w$
|
||||
bereits in $C$ gewesen -- die Bedingung „beide Endpunkte frei'' wäre verletzt
|
||||
und die Kante nicht aufgenommen worden. Also sind die Kanten aus $A$ paarweise
|
||||
knotendisjunkt.
|
||||
|
||||
\textbf{Schranke für das Optimum.} $C^\ast$ deckt jede Kante aus $A$ mit
|
||||
mindestens einem Endpunkt ab. Da die Kanten aus $A$ knotendisjunkt sind,
|
||||
braucht jede einen \emph{eigenen} Knoten in $C^\ast$, also $|C^\ast| \ge |A|$.
|
||||
|
||||
\textbf{Kombination.}
|
||||
\[
|
||||
|C| = 2|A| \le 2\,|C^\ast|.
|
||||
\]
|
||||
Somit hat \problem{2ApproxVC} die Güte $2$. \hfill$\square$
|
||||
|
||||
\textbf{c)} Der Kreis $C_n$ (mit $n$ gerade) hat als größte unabhängige Menge
|
||||
die $n/2$ „jeden zweiten Knoten''; also ist die maximale Unabhängigkeitszahl
|
||||
$n/2$ und ein minimales Vertex Cover hat Größe
|
||||
\[
|
||||
|C^\ast| = n - \frac{n}{2} = \frac{n}{2}.
|
||||
\]
|
||||
\emph{Schlechteste Kantenreihenfolge:} Man ordnet die $n/2$ paarweise
|
||||
disjunkten Kanten
|
||||
$\{v_1,v_2\}, \{v_3,v_4\}, \dots, \{v_{n-1},v_n\}$ zuerst an (danach beliebig
|
||||
die restlichen Kanten). Beim Abarbeiten dieser $n/2$ Kanten sind jeweils beide
|
||||
Endpunkte noch frei, sodass der Algorithmus \emph{alle} $n$ Knoten aufnimmt;
|
||||
die übrigen Kanten haben dann stets einen bereits bedeckten Endpunkt. Es folgt
|
||||
$|C| = n$ und die Rate
|
||||
\[
|
||||
\frac{|C|}{|C^\ast|} = \frac{n}{n/2} = 2,
|
||||
\]
|
||||
also der nach b) größtmögliche Wert.
|
||||
|
||||
\textbf{d) BONUS.} Wir geben die Familie der \emph{ungeraden} Kreise
|
||||
$C_{2k+1}$ für $k \ge 2$ an, mit Knoten $v_1, \dots, v_{2k+1}$ und Kanten
|
||||
$\{v_i, v_{i+1}\}$ (Indizes modulo $2k+1$).
|
||||
\begin{itemize}
|
||||
\item \emph{Optimum:} Die größte unabhängige Menge in $C_{2k+1}$ hat Größe $k$
|
||||
(man kann höchstens jeden zweiten Knoten wählen, im ungeraden Kreis also $k$).
|
||||
Damit ist $|C^\ast| = (2k+1) - k = k+1$.
|
||||
\item \emph{Ungünstige Reihenfolge:} Ordne die $k$ paarweise disjunkten Kanten
|
||||
$\{v_1,v_2\}, \{v_3,v_4\}, \dots, \{v_{2k-1},v_{2k}\}$ zuerst an. Beim
|
||||
Abarbeiten sind jeweils beide Endpunkte frei, also nimmt der Algorithmus die
|
||||
$2k$ Knoten $v_1, \dots, v_{2k}$ auf. Der einzige unbedeckte Knoten wäre
|
||||
$v_{2k+1}$; seine beiden Kanten $\{v_{2k}, v_{2k+1}\}$ und
|
||||
$\{v_{2k+1}, v_1\}$ haben aber je einen Endpunkt in $C$, werden also
|
||||
übersprungen. Somit $|C| = 2k$.
|
||||
\item \emph{Rate:}
|
||||
\[
|
||||
\frac{|C|}{|C^\ast|} = \frac{2k}{k+1} \;=\; 2 - \frac{2}{k+1}
|
||||
\;\xrightarrow[k \to \infty]{}\; 2,
|
||||
\]
|
||||
und die Rate ist streng monoton wachsend sowie für jedes endliche $k$ echt
|
||||
kleiner als $2$ (z.\,B.\ $C_5{:}\ \tfrac43$, $C_7{:}\ \tfrac32$,
|
||||
$C_9{:}\ \tfrac85$, $C_{11}{:}\ \tfrac53$).
|
||||
\end{itemize}
|
||||
Die Approximationsrate von \problem{2ApproxVC} konvergiert also entlang der
|
||||
Familie $(C_{2k+1})_{k \ge 2}$ gegen $2$; zusammen mit b) ist die Güte $2$
|
||||
damit asymptotisch scharf. \hfill$\square$
|
||||
|
||||
\end{document}
|
||||
33
lernmaterial/muster.aux
Normal file
33
lernmaterial/muster.aux
Normal file
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|
||||
\relax
|
||||
\providecommand\hyper@newdestlabel[2]{}
|
||||
\providecommand\HyField@AuxAddToFields[1]{}
|
||||
\providecommand\HyField@AuxAddToCoFields[2]{}
|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
242
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Normal file
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Normal file
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[48{/usr/share/texmf/fonts/enc/dvips/lm/lm-rmtt.enc}] (./muster.aux)
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||||
***********
|
||||
LaTeX2e <2023-11-01> patch level 1
|
||||
L3 programming layer <2024-01-22>
|
||||
***********
|
||||
|
||||
|
||||
LaTeX Font Warning: Some font shapes were not available, defaults substituted.
|
||||
|
||||
Package rerunfilecheck Info: File `muster.out' has not changed.
|
||||
(rerunfilecheck) Checksum: 87A182F17260A6E4F439DC4972AA0E25;939.
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
24578 strings out of 476182
|
||||
443005 string characters out of 5795595
|
||||
1950975 words of memory out of 5000000
|
||||
45700 multiletter control sequences out of 15000+600000
|
||||
695002 words of font info for 292 fonts, out of 8000000 for 9000
|
||||
14 hyphenation exceptions out of 8191
|
||||
107i,9n,110p,1003b,807s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/share/texmf/fonts/type1/public/lm/lmbx10.pfb></usr/share/texmf/fonts/ty
|
||||
pe1/public/lm/lmbx12.pfb></usr/share/texmf/fonts/type1/public/lm/lmcsc10.pfb></
|
||||
usr/share/texmf/fonts/type1/public/lm/lmex10.pfb></usr/share/texmf/fonts/type1/
|
||||
public/lm/lmmi10.pfb></usr/share/texmf/fonts/type1/public/lm/lmmi6.pfb></usr/sh
|
||||
are/texmf/fonts/type1/public/lm/lmmi7.pfb></usr/share/texmf/fonts/type1/public/
|
||||
lm/lmmi8.pfb></usr/share/texmf/fonts/type1/public/lm/lmmi9.pfb></usr/share/texm
|
||||
f/fonts/type1/public/lm/lmr10.pfb></usr/share/texmf/fonts/type1/public/lm/lmr12
|
||||
.pfb></usr/share/texmf/fonts/type1/public/lm/lmr6.pfb></usr/share/texmf/fonts/t
|
||||
ype1/public/lm/lmr7.pfb></usr/share/texmf/fonts/type1/public/lm/lmr8.pfb></usr/
|
||||
share/texmf/fonts/type1/public/lm/lmr9.pfb></usr/share/texmf/fonts/type1/public
|
||||
/lm/lmri10.pfb></usr/share/texmf/fonts/type1/public/lm/lmss10.pfb></usr/share/t
|
||||
exmf/fonts/type1/public/lm/lmsy10.pfb></usr/share/texmf/fonts/type1/public/lm/l
|
||||
msy5.pfb></usr/share/texmf/fonts/type1/public/lm/lmsy6.pfb></usr/share/texmf/fo
|
||||
nts/type1/public/lm/lmsy7.pfb></usr/share/texmf/fonts/type1/public/lm/lmsy8.pfb
|
||||
></usr/share/texmf/fonts/type1/public/lm/lmsy9.pfb></usr/share/texmf/fonts/type
|
||||
1/public/lm/lmtt10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
|
||||
ts/symbols/msam10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfont
|
||||
s/symbols/msbm10.pfb>
|
||||
Output written on muster.pdf (48 pages, 572020 bytes).
|
||||
PDF statistics:
|
||||
427 PDF objects out of 1000 (max. 8388607)
|
||||
317 compressed objects within 4 object streams
|
||||
80 named destinations out of 1000 (max. 500000)
|
||||
77349 words of extra memory for PDF output out of 89155 (max. 10000000)
|
||||
|
||||
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