init

aufgaben.md

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+Anwendung
+- Greedy anwenden
+- ModifiedGreedy anwenden 
+- Sahni anwenden 
+- ListScheduling anwenden 
+- LPT anwenden 
+- RoundRobin anwenden 
+- Parallel-Task-Scheduling anwenden 
+- TSP1 anwenden 
+- Christofides anwenden 
+- Strip Packing: NFDH anwenden 
+
+Vorlesungsbeweis
+- P ⊆ NP zeigen
+- NDTM-Definition ⊆ Verifizierer-Definition zeigen
+- Verifizierer-Definition ⊆ NDTM-Definition zeigen
+- Transitivität von Polynomialzeitreduktionen zeigen
+- Vererbung der NP-Vollständigkeit zeigen
+- L NP-vollständig: L in P <=> P = NP zeigen
+
+NP-Vollständigkeit
+- NP-Vollständigkeit zeigen für k-Clique
+- NP-Vollständigkeit zeigen für 3-SAT
+- NP-Vollständigkeit zeigen für VertexCover
+- NP-Vollständigkeit zeigen für CliqueAndIndependentSet
+- NP-Vollständigkeit zeigen für k-CliqueUniversal
+- NP-Vollständigkeit zeigen für Clique-Nomember
+- NP-Vollständigkeit zeigen für k-COLOR-PRECOLORING
+- NP-Vollständigkeit zeigen für FeedbackVertexSet
+- NP-Vollständigkeit zeigen für Delta-Cover
+- NP-Vollständigkeit zeigen für 3-COLOR mit Minimalgrad 3
+- NP-Vollständigkeit zeigen für HamiltonianPath
+- NP-Vollständigkeit zeigen für Hitchhiker's-HamiltonianCycle
+- NP-Vollständigkeit zeigen für TSP-Entscheidung
+- NP-Vollständigkeit zeigen für k-CLIQUE-DEG-3
+- NP-Vollständigkeit zeigen für Partition
+- NP-Vollständigkeit zeigen für SubsetSumCardinality
+- NP-Vollständigkeit zeigen für (a1=1)-SubsetSum
+- NP-Vollständigkeit zeigen für SubsetSum mit Teilbarkeit
+- NP-Vollständigkeit zeigen für SubsetSum ohne Zweierpotenzen
+- NP-Vollständigkeit zeigen für AtMostTwoPerSize-SubsetSum
+- NP-Schwere zeigen für HALT_TM
+
+Approximative Algorithmen
+- Güte zeigen für ListScheduling
+- Güte zeigen für LPT
+- Güte zeigen für RoundRobin
+- Güte zeigen für Parallel-Task-ListScheduling
+- Güte zeigen für ModifiedGreedy
+- Güte zeigen für Sahni
+- Güte wiederlegen für Greedy
+- Güte zeigen für TSP1
+- Güte zeigen für Christofides
+- Güte zeigen für 2ApproxVC
+- Güte zeigen für MAX-3-SAT
+- Güte zeigen für ApproximateSubsetSum
+- Güte zeigen für Strip Packing: NFDH
+
+ETH
+- ETH-Schranke zeigen für VertexCover
+- ETH-Schranke zeigen für HittingSet
+- ETH-Schranke zeigen für k-Clique
+- ETH-Schranke zeigen für k-IndependentSet
+- ETH-Schranke zeigen für 3-DM
+- ETH-Schranke zeigen für 3-ExactCover
+- ETH-Schranke zeigen für SubsetSum
+- ETH-Schranke zeigen für k-Color
+- ETH-Schranke zeigen für CoverClique
+- ETH-Schranke zeigen für Delta-Cover
+- ETH-Schranke zeigen für NAE-4-SAT
+- ETH-Schranke zeigen für SetSplitting
+- ETH-Schranke zeigen für DominatingSet
+- ETH-Schranke zeigen für GridTiling
+- ETH-Schranke zeigen für SetCover
+- ETH-Schranke zeigen für Scheduling 2|prec, p in {1,2}|Cmax
+- ETH-Schranke zeigen für ILP-Feasibility

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hausaufgaben/meine_loeasung/hausaufgabe_10_loesung.tex

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+\documentclass[12pt]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[ngerman,provide=*]{babel}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[a4paper,margin=2.5cm]{geometry}
+
+\title{Hausaufgabe 10}
+\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
+\date{21.\ Juni 2026}
+
+\begin{document}
+
+\maketitle
+
+\section*{Hausaufgabe 10.1}
+
+\subsection*{Teil 1: FeedbackVertexSet ist in NP}
+
+$FEEDBACKVERTEXSET \in NP$ wurde in Hausaufgabe 9.2 gezeigt. 
+
+\subsection*{Teil 2: FeedbackVertexSet ist NP-schwer}
+
+Zeige, dass $FEEDBACKVERTEXSET$ NP-schwer ist durch die Reduktion: \\
+$VERTEXCOVER \le FEEDBACKVERTEXSET$. \\
+
+
+\textbf{Konstruktion}: Sei $G$ und $k$ eine Instanz von $VERTEXCOVER$. \\
+Wir konstruieren daraus ein $FEEDBACKVERTEXSET$ durch:
+\begin{itemize}
+  \item $V' = V$
+  \item $E' = \{ (u,v), (v,u) \mid \{u,v\} \in E \}$
+  \item $k' = k$
+\end{itemize}
+
+\textbf{Beweis VertexCover nach FeedbackVertexSet}:
+
+\begin{itemize}
+  \item Sei $VC$ ein VertexCover von $G$ mit $|VC| \le k$.
+  \item Somit hat jede Kante aus $G$ einen Endpunkt in $VC$.
+  \item Also hat $G \setminus VC$ keine Kanten mehr.
+  \item Dann hat auch $G' \setminus VC$ keine Kanten und damit keine Kreise.
+  \item Also ist $VC$ ein FeedbackVertexSet für $G'$ mit $|VC| \le k'$.
+  \end{itemize}
+
+\textbf{Beweis FeedbackVertexSet nach VertexCover}:
+
+\begin{itemize}
+  \item Sei $FVS$ ein FeedbackVertexSet von $G'$ mit $|FVS| \le k'$.
+  \item Dann ist $G' \setminus FVS$ kreisfrei.
+  \item Angenommen $FVS$ wäre kein VertexCover. 
+  \item Dann gäbe es eine Kante $\{u, v\} \in E$ mit $u, v \notin FVS$.
+  \item Somit wären $u, v$ noch in $G' \setminus FVS$ vorhanden.
+  \item Die Kanten $(u, v)$ und $(v, u)$ würden einen Kreis erzeugen.
+  \item Das wäre ein Widerspruch zu $G' \setminus FVS$ ist kreisfrei.
+  \item Also ist $FVS$ ein VertexCover von $G$ mit $|FVS| \le k$.
+\end{itemize}
+
+\textbf{Laufzeit}: Nur die Kanten werden verdoppelt $O(|E|)$,
+also bleibt die Transformation polynomiell. \\
+
+
+Da $FEEDBACKVERTEXSET \in NP$ und NP-schwer ist, folgt dass
+$FEEDBACKVERTEXSET$ NP-vollständig ist. \hfill $\square$
+
+\newpage
+
+\section*{Hausaufgabe 10.2}
+
+\subsection*{Teil 1: $k$-CliqueUniversal ist in NP}
+
+Eingabe: Graph $G = (V, E)$, Zahl $k$ und Zertifikat $C \subseteq V$. \\
+
+Der Verifizierer arbeitet wie folgt:
+
+\begin{itemize}
+  \item Prüfe ob $|C| \ge k$, lehne ab falls $|C| < k$.
+  \item Prüfe für jedes Paar $\{x, y\} \subseteq C$ ob $\{x, y\} \in E$.
+  \item Lehne ab, falls für ein Paar $\{x, y\} \notin E$.
+  \item Sonst akzeptiere.
+\end{itemize}
+
+Existiert eine Clique $C$ mit $|C| \ge k$, dann gibt es ein Zertifikat $C$
+dafür. Dieses wird vom Verifizierer akzeptiert und sonst abgelehnt. \\
+
+Prüfung der Größe und aller Paare läuft in $O(|V|^2)$, somit in poly. Zeit.
+Somit ist das ein poly. Verifizierer.
+Also gilt $k\text{-}CLIQUEUNIVERSAL \in NP$.
+
+\subsection*{Teil 2: $k$-CliqueUniversal ist NP-schwer}
+
+Zeige, dass $k\text{-}CLIQUEUNIVERSAL$ NP-schwer ist durch die Reduktion: \\
+$CLIQUE \le k\text{-}CLIQUEUNIVERSAL$. \\
+$CLIQUE$ ist bereits als NP-vollständig bekannt. \\
+
+\textbf{Konstruktion}: Sei $G$ und $k$ eine Instanz von $CLIQUE$. \\
+Wir konstruieren daraus ein $k\text{-}CLIQUEUNIVERSAL$ durch:
+\begin{itemize}
+  \item $V' = V \cup \{u\}$ mit einem neuen Knoten $u$
+  \item $E' = E \cup \{ \{u, v\} \mid v \in V \}$
+  \item $k' = k + 1$
+\end{itemize}
+
+\textbf{Beweis Clique nach $k$-CliqueUniversal}:
+
+\begin{itemize}
+  \item Sei $C$ eine Clique von $G$ mit $|C| \ge k$.
+  \item Da $u$ mit allen Knoten verbunden ist, ist $u$ auch mit allen
+        Knoten aus $C$ verbunden.
+  \item Damit ist $C \cup \{u\}$ eine Clique in $G'$.
+  \item Dann gilt auch $|C \cup \{u\}| \ge k + 1 = k'$.
+\end{itemize}
+
+\textbf{Beweis $k$-CliqueUniversal nach Clique}:
+
+\begin{itemize}
+  \item Sei $C'$ eine Clique von $G'$ mit $|C'| \ge k'$.
+  \item Setze $C = C' \setminus \{u\}$, dann gilt $|C| \ge k' - 1 = k$.
+  \item Alle Knoten in $C$ liegen in $V$ und alle Kanten zwischen ihnen
+        liegen in $E$, da nur Kanten zu $u$ neu hinzugefügt wurden.
+  \item Damit ist $C$ eine Clique in $G$ mit $|C| \ge k$.
+\end{itemize}
+
+\textbf{Laufzeit}: Es werden ein Knoten und $|V|$ Kanten hinzugefügt $O(|V|)$,
+also bleibt die Transformation polynomiell. \\
+
+
+Da $k\text{-}CLIQUEUNIVERSAL \in NP$ und NP-schwer ist, folgt dass
+$k\text{-}CLIQUEUNIVERSAL$ NP-vollständig ist. \hfill $\square$
+
+\end{document}

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hausaufgaben/meine_loeasung/hausaufgabe_11_loesung.tex

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+\documentclass[12pt]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[ngerman,provide=*]{babel}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[a4paper,margin=2.5cm]{geometry}
+
+\title{Hausaufgabe 11}
+\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
+\date{29.\ Juni 2026}
+
+\begin{document}
+
+\maketitle
+
+\section*{Hausaufgabe 11.1}
+
+\subsection*{Teil 1: $\Delta$-Cover ist in NP}
+
+Eingabe: Graph $G = (V, E)$, Zahl $k$ und Zertifikat $C_\Delta \subseteq V$. \\
+
+Der Verifizierer arbeitet wie folgt:
+
+\begin{itemize}
+  \item Prüfe ob $|C_\Delta| \le k$, lehne ab falls $|C_\Delta| > k$.
+  \item Prüfe für jedes Knotentripel $\{a, b, c\} \subseteq V$, ob es ein
+        Dreieck bildet (also ob $\{a,b\}, \{b,c\}, \{a,c\} \in E$).
+  \item Lehne ab, falls ein solches Dreieck kein $a$, $b$ oder $c$ in
+        $C_\Delta$ hat.
+  \item Sonst akzeptiere.
+\end{itemize}
+
+Existiert eine Dreiecksüberdeckung $C_\Delta$ mit $|C_\Delta| \le k$, dann gibt
+es ein Zertifikat dafür. Dieses wird vom Verifizierer akzeptiert und sonst
+abgelehnt. \\
+
+Die Größenprüfung läuft in $O(|V|)$ und das Prüfen aller Tripel in $O(|V|^3)$,
+somit in poly. Zeit. Also gilt $\Delta\text{-}COVER \in NP$.
+
+\subsection*{Teil 2: $\Delta$-Cover ist NP-schwer}
+
+Zeige, dass $\Delta\text{-}COVER$ NP-schwer ist durch die Reduktion: \\
+$VERTEXCOVER \le \Delta\text{-}COVER$. \\
+$VERTEXCOVER$ ist bereits als NP-vollständig bekannt. \\
+
+\textbf{Konstruktion}: Sei $G = (V, E)$ und $k$ eine Instanz von $VERTEXCOVER$. \\
+Wir konstruieren daraus eine $\Delta\text{-}COVER$-Instanz $G' = (V', E')$ durch:
+\begin{itemize}
+  \item Für jede Kante $e = \{u, v\} \in E$ füge einen neuen Knoten $w_e$ hinzu.
+  \item $V' = V \cup \{ w_e \mid e \in E \}$
+  \item $E' = E \cup \{ \{u, w_e\}, \{v, w_e\} \mid e = \{u,v\} \in E \}$
+  \item $k' = k$
+\end{itemize}
+So wird aus jeder Kante $e = \{u, v\}$ ein Dreieck $\{u, v, w_e\}$. \\
+
+\textbf{Beweis VertexCover nach $\Delta$-Cover}:
+
+\begin{itemize}
+  \item Sei $C$ ein VertexCover von $G$ mit $|C| \le k$.
+  \item Zu zeigen sei, dass $C$ auch eine $\Delta$-Cover für $G'$ mit
+        $|C| \le k'$ ist.
+  \item Jedes Dreieck in $G'$ hat die Form $\{u, v, w_e\}$ für eine Kante
+        $e = \{u, v\} \in E$.
+  \item Da $C$ ein VertexCover ist, gilt $u \in C$ oder $v \in C$.
+  \item Somit hat jedes Dreieck mindestens einen Knoten in $C$.
+  \item Also ist $C$ eine $\Delta$-Cover für $G'$ mit $|C| \le k'$.
+\end{itemize}
+
+\textbf{Beweis $\Delta$-Cover nach VertexCover}:
+
+\begin{itemize}
+  \item Sei $C_\Delta$ eine $\Delta$-Cover für $G'$ mit $|C_\Delta| \le k'$.
+  \item Jede Kante $e = \{u, v\} \in E$ bildet durch $\{u, v, w_e\}$ ein Dreieck
+        in $G'$.
+  \item Also gilt $u, v$ oder $w_e \in C_\Delta$ für alle Kanten $e \in E$.
+  \item Falls $w_e \in C_\Delta$, dann kann man $w_e$ durch $u$ ersetzen, ohne
+        die Eigenschaft von $\Delta$-Cover zu verletzen.
+  \item Denn $w_e$ liegt in genau einem Dreieck $\{u, v, w_e\}$, das auch durch
+        $u$ gedeckt wird.
+  \item So entsteht eine Menge $C$ mit $|C| \le |C_\Delta| \le k$, die nur noch
+        Knoten aus $V$ enthält.
+  \item Für jede Kante $e = \{u, v\} \in E$ gilt nun $u \in C$ oder $v \in C$.
+  \item Also ist $C$ ein VertexCover für $G$ mit $|C| \le k$.
+\end{itemize}
+
+\textbf{Laufzeit}: Pro Kante werden ein Knoten und zwei Kanten hinzugefügt
+$O(|E|)$, also bleibt die Transformation polynomiell. \\
+
+
+Da $\Delta\text{-}COVER \in NP$ und NP-schwer ist, folgt dass
+$\Delta\text{-}COVER$ NP-vollständig ist. \hfill $\square$
+
+\newpage
+
+\section*{Hausaufgabe 11.2}
+
+\subsection*{$HALT_{TM}$ ist NP-schwer}
+
+Zeige, dass $HALT_{TM}$ NP-schwer ist durch die Reduktion: \\
+$SAT \le HALT_{TM}$. \\
+$SAT$ ist bereits als NP-vollständig bekannt, also insbesondere NP-schwer. \\
+
+\textbf{Konstruktion}: Sei $\varphi$ eine $SAT$-Instanz mit den Variablen
+$x_1, \dots, x_n$. \\
+Wir konstruieren daraus eine $HALT_{TM}$-Instanz $\langle M, w \rangle$ durch:
+\begin{itemize}
+  \item $w = \langle \varphi \rangle$, also die Kodierung der Formel $\varphi$.
+  \item $M$ ist eine feste DTM , die auf Eingabe
+        $\langle \varphi \rangle$ wie folgt arbeitet:
+  \begin{enumerate}
+    \item Lese $\varphi$ und probiere nacheinander alle $2^n$ Belegungen
+          $\beta$ der Variablen $x_1, \dots, x_n$.
+    \item Falls eine Belegung $\beta$ die Formel $\varphi$ erfüllt, halte.
+    \item Falls alle $2^n$ Belegungen geprüft wurden und keine $\varphi$
+          erfüllt, gehe in eine Endlosschleife.
+  \end{enumerate}
+\end{itemize}
+Die Abbildung $f(\varphi) = \langle M, w \rangle$ kodiert nur $\varphi$; die
+Maschine $M$ ist konstant. Somit ist $f$ in poly. Zeit berechenbar. \\
+
+\textbf{Beweis SAT nach $HALT_{TM}$}:
+
+\begin{itemize}
+  \item Sei $\varphi \in SAT$.
+  \item Dann existiert eine erfüllende Belegung $\beta$.
+  \item $M$ probiert alle Belegungen und findet $\beta$, also hält $M$ auf $w$.
+  \item Damit gilt $\langle M, w \rangle \in HALT_{TM}$.
+\end{itemize}
+
+\textbf{Beweis $HALT_{TM}$ nach SAT}:
+
+\begin{itemize}
+  \item Sei $\langle M, w \rangle \in HALT_{TM}$, also hält $M$ auf $w$.
+  \item $M$ hält nur, wenn es eine erfüllende Belegung $\beta$ findet.
+  \item Denn ohne erfüllende Belegung geht $M$ in die Endlosschleife.
+  \item Also existiert eine erfüllende Belegung für $\varphi$.
+  \item Damit gilt $\varphi \in SAT$.
+\end{itemize}
+
+\textbf{Laufzeit}: Es wird nur $\varphi$ kodiert und die feste Maschine $M$
+ausgegeben $O(|\varphi|)$, also bleibt die Transformation polynomiell. \\
+
+
+Da $SAT$ NP-schwer ist und $SAT \le HALT_{TM}$ gilt, folgt dass $HALT_{TM}$
+NP-schwer ist. \hfill $\square$
+
+\end{document}

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hausaufgaben/meine_loeasung/hausaufgabe_12_loesung.tex

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+\documentclass[12pt]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[ngerman,provide=*]{babel}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[a4paper,margin=2.5cm]{geometry}
+
+\title{Hausaufgabe 12}
+\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
+\date{6.\ Juli 2026}
+
+\begin{document}
+
+\maketitle
+
+\section*{Hausaufgabe 12.1}
+
+\subsection*{Teil (a): Der Algorithmus hat Güte 2}
+
+Zu zeigen ist, dass der Algorithmus die Güte 2 hat, also die Lösung vom
+Algorithmus mindestens halb so gut ist wie die optimale Lösung. Also
+$\nu(A(\varphi)) \ge \frac{1}{2}\nu(OPT(\varphi))$ für alle Eingaben $\varphi$. \\
+
+Sei $\varphi$ eine beliebige 3-SAT-Formel mit $m$ Klauseln
+$C_1, C_2, \dots, C_m$, wobei jede Klausel genau 3 Literale enthält. \\
+
+Betrachte die beiden Belegungen $\beta_0$ (alle Variablen false) und $\beta_1$
+(alle Variablen true):
+\begin{itemize}
+  \item Unter $\beta_0$ ist jedes negative Literal wahr.
+  \item Unter $\beta_1$ ist jedes positive Literal wahr.
+\end{itemize}
+Jedes Literal ist positiv oder negativ. Jede Klausel $C_i$ enthält mindestens
+ein Literal. Also wird jede Klausel unter $\beta_0$ oder unter $\beta_1$ (oder
+unter beiden) erfüllt. \\
+
+Sei $\nu_0 = \nu(\beta_0)$ die Anzahl der von $\beta_0$ erfüllten Klauseln und
+$\nu_1 = \nu(\beta_1)$ die Anzahl der von $\beta_1$ erfüllten Klauseln. Da jede
+Klausel zu $\nu_0$ oder $\nu_1$ zählt, gilt
+\[
+  \nu_0 + \nu_1 \ge m.
+\]
+
+Der Algorithmus wählt die Belegung mit den meisten erfüllten Klauseln, also
+\[
+  \nu(A(\varphi)) = \max\{\nu_0, \nu_1\} \ge \frac{\nu_0 + \nu_1}{2}
+  \ge \frac{m}{2}.
+\]
+
+Da die optimale Lösung höchstens alle $m$ Klauseln erfüllen kann, gilt
+$\nu(OPT(\varphi)) \le m$. Daraus folgt
+\[
+  \nu(A(\varphi)) \ge \frac{m}{2} \ge \frac{\nu(OPT(\varphi))}{2}.
+\]
+
+Somit hat der Algorithmus eine Güte von 2. \hfill $\square$
+
+\subsection*{Teil (b): Formel mit genau der Hälfte}
+
+Der Algorithmus erfüllt genau die Hälfte der maximal erfüllbaren Klauseln, wenn
+$\beta_0$ und $\beta_1$ jeweils nur eine Klausel erfüllen, das Optimum aber
+beide Klauseln erfüllt. \\
+
+\textbf{Formel:}
+$\varphi = (x_1 \vee x_2 \vee x_3) \wedge (\neg x_1 \vee \neg x_2 \vee \neg x_3)$
+
+\textbf{Unter $\beta_0$ (alle false):}
+\begin{itemize}
+  \item $C_1 = (x_1 \vee x_2 \vee x_3) = (\text{false} \vee \text{false} \vee \text{false}) = \text{false}$
+  \item $C_2 = (\neg x_1 \vee \neg x_2 \vee \neg x_3) = (\text{true} \vee \text{true} \vee \text{true}) = \text{true}$
+  \item $\nu(\beta_0) = 1$
+\end{itemize}
+
+\textbf{Unter $\beta_1$ (alle true):}
+\begin{itemize}
+  \item $C_1 = (x_1 \vee x_2 \vee x_3) = (\text{true} \vee \text{true} \vee \text{true}) = \text{true}$
+  \item $C_2 = (\neg x_1 \vee \neg x_2 \vee \neg x_3) = (\text{false} \vee \text{false} \vee \text{false}) = \text{false}$
+  \item $\nu(\beta_1) = 1$
+\end{itemize}
+
+Der Algorithmus wählt
+$\nu(A(\varphi)) = \max\{\nu(\beta_0), \nu(\beta_1)\} = \max\{1, 1\} = 1$. \\
+
+Eine optimale Lösung wäre zum Beispiel $x_1 = \text{true}, x_2 = \text{false},
+x_3 = \text{false}$:
+\begin{itemize}
+  \item $C_1 = (\text{true} \vee \text{false} \vee \text{false}) = \text{true}$
+  \item $C_2 = (\text{false} \vee \text{true} \vee \text{true}) = \text{true}$
+\end{itemize}
+
+Daraus ergibt sich $\nu(OPT(\varphi)) = 2$, also
+$\nu(A(\varphi)) = 1 = \frac{\nu(OPT(\varphi))}{2}$. Der Algorithmus erfüllt
+genau die Hälfte der maximal erfüllbaren Klauseln. \hfill $\square$
+
+\end{document}

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hausaufgaben/meine_loeasung/hausaufgabe_9_loesung.tex

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+\documentclass[12pt]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[ngerman,provide=*]{babel}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[a4paper,margin=2.5cm]{geometry}
+
+\title{Hausaufgabe 9}
+\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
+\date{15.\ Juni 2026}
+
+\begin{document}
+
+\maketitle
+
+\section*{Hausaufgabe 9.1}
+
+\subsection*{Teil 1}
+
+Ein minimales Vertex Cover ist $C = \{2, 7, 5, 4, 1\}$.
+
+\subsection*{Teil 2}
+
+Der Algorithmus arbeitet wie folgt:
+
+\begin{itemize}
+  \item Finde nichtdeterministisch die Knoten für $C$. 
+  \item Prüfe ob $|C| \le k$, lehne ab falls $|C| > k$.
+  \item Prüfe für jede Kante $\{u, v\} \in E$ ob $v \in C$ oder $u \in C$.
+  \item Lehne ab, falls für eine Kante $u, v \notin C$.
+  \item Sonst akzeptiere.
+\end{itemize}
+
+Existiert ein Vertex Cover kleiner-gleich $k$, 
+dann gibt es auch eine Knotenmenge $|C| \le k$ dafür. 
+Diese wird vom Algorithmus gefunden und akzeptiert. 
+Falls es keine solche Knotenmenge gibt, dann lehnt er ab. \\
+
+Auswahl der Knoten und Prüfung der Kanten läuft in $O(|V| + |E|)$, somit in poly. Zeit.
+Somit ist das eine poly. NDTM.
+Also gilt $VERTEXCOVER \in NP$. 
+
+\newpage
+
+\section*{Hausaufgabe 9.2}
+
+\subsection*{Teil 1}
+
+Ein minimales Feedback Vertex Set ist $X = \{1, 5\}$.
+
+\subsection*{Teil 2}
+
+Eingabe: Graph $G=(V,E)$ und Zertifikat $X \subseteq V$. \\
+
+Der Verifizierer arbeitet wie folgt:
+
+\begin{itemize}
+  \item Prüfe ob $|X| \le k$, lehne ab falls $|X| > k$. 
+  \item Konstruiere $H=(V \setminus X,E)$. 
+  \item Starte eine Tiefensuche auf $H$ und markiere bereits besuchte Knoten grau.
+  \item Beim Backtracking markiere Knoten schwarz. 
+  \item Wird ein grauer Knoten aufgerufen, dann lehne ab. 
+  \item Sonst akzeptiere.
+
+\end{itemize}
+
+Existiert ein Feedback Vertex Set kleiner-gleich $k$, 
+dann gibt es ein Zertifikat $X$ mit $|X| \le k$ dafür. 
+Die Knotenmenge wird vom Verifizierer akzeptiert und sonst abgelehnt. \\ 
+
+Prüfung von $X$, Konstruktion von $H$ und die Tiefensuche laufen in $O(|V| + |E|)$. 
+Somit ist das ein poly. Verifizierer.
+Also gilt $FEEDBACKVERTEXSET \in NP$. 
+
+\end{document}

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lernmaterial/aufgaben.toc

@@ -0,0 +1,72 @@
+\contentsline {section}{\numberline {1}Anwendung}{2}{}%
+\contentsline {subsection}{1\ \ Greedy anwenden}{3}{}%
+\contentsline {subsection}{2\ \ ModifiedGreedy anwenden}{4}{}%
+\contentsline {subsection}{3\ \ Sahni anwenden}{5}{}%
+\contentsline {subsection}{4\ \ ListScheduling anwenden}{6}{}%
+\contentsline {subsection}{5\ \ LPT anwenden}{7}{}%
+\contentsline {subsection}{6\ \ RoundRobin anwenden}{8}{}%
+\contentsline {subsection}{7\ \ Parallel-Task-Scheduling anwenden}{9}{}%
+\contentsline {subsection}{8\ \ TSP1 anwenden}{10}{}%
+\contentsline {subsection}{9\ \ Christofides anwenden}{11}{}%
+\contentsline {subsection}{10\ \ Strip Packing: NFDH anwenden}{12}{}%
+\contentsline {section}{\numberline {2}Vorlesungsbeweis}{13}{}%
+\contentsline {subsection}{11\ \ $P \subseteq \textsf {NP}$ zeigen}{14}{}%
+\contentsline {subsection}{12\ \ NDTM-Definition $\subseteq $ Verifizierer-Definition zeigen}{15}{}%
+\contentsline {subsection}{13\ \ Verifizierer-Definition $\subseteq $ NDTM-Definition zeigen}{16}{}%
+\contentsline {subsection}{14\ \ Transitivität von Polynomialzeitreduktionen zeigen}{17}{}%
+\contentsline {subsection}{15\ \ Vererbung der NP-Vollständigkeit zeigen}{18}{}%
+\contentsline {subsection}{16\ \ $L$ NP-vollständig: $L \in P \Leftrightarrow P = \textsf {NP}$ zeigen}{19}{}%
+\contentsline {section}{\numberline {3}NP-Vollständigkeit}{20}{}%
+\contentsline {subsection}{17\ \ NP-Vollständigkeit zeigen für $k$-Clique}{21}{}%
+\contentsline {subsection}{18\ \ NP-Vollständigkeit zeigen für 3-SAT}{23}{}%
+\contentsline {subsection}{19\ \ NP-Vollständigkeit zeigen für VertexCover}{24}{}%
+\contentsline {subsection}{20\ \ NP-Vollständigkeit zeigen für CliqueAndIndependentSet}{25}{}%
+\contentsline {subsection}{21\ \ NP-Vollständigkeit zeigen für $k$-CliqueUniversal}{27}{}%
+\contentsline {subsection}{22\ \ NP-Vollständigkeit zeigen für Clique-Nomember}{28}{}%
+\contentsline {subsection}{23\ \ NP-Vollständigkeit zeigen für $k$-COLOR-PRECOLORING}{29}{}%
+\contentsline {subsection}{24\ \ NP-Vollständigkeit zeigen für FeedbackVertexSet}{30}{}%
+\contentsline {subsection}{25\ \ NP-Vollständigkeit zeigen für $\Delta $-Cover}{32}{}%
+\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}{}%
+\contentsline {subsection}{39\ \ Güte zeigen für LPT}{53}{}%
+\contentsline {subsection}{40\ \ Güte zeigen für RoundRobin}{54}{}%
+\contentsline {subsection}{41\ \ Güte zeigen für Parallel-Task-ListScheduling}{55}{}%
+\contentsline {subsection}{42\ \ Güte zeigen für ModifiedGreedy}{56}{}%
+\contentsline {subsection}{43\ \ Güte zeigen für Sahni}{57}{}%
+\contentsline {subsection}{44\ \ Güte widerlegen für Greedy}{58}{}%
+\contentsline {subsection}{45\ \ Güte zeigen für $\Delta $TSP1}{59}{}%
+\contentsline {subsection}{46\ \ Güte zeigen für Christofides}{60}{}%
+\contentsline {subsection}{47\ \ Güte zeigen für 2ApproxVC}{61}{}%
+\contentsline {subsection}{48\ \ Güte zeigen für MAX-3-SAT}{62}{}%
+\contentsline {subsection}{49\ \ Güte zeigen für ApproximateSubsetSum}{63}{}%
+\contentsline {subsection}{50\ \ Güte zeigen für Strip Packing: NFDH}{64}{}%
+\contentsline {section}{\numberline {5}ETH}{65}{}%
+\contentsline {subsection}{51\ \ ETH-Schranke zeigen für VertexCover}{66}{}%
+\contentsline {subsection}{52\ \ ETH-Schranke zeigen für HittingSet}{67}{}%
+\contentsline {subsection}{53\ \ ETH-Schranke zeigen für $k$-Clique}{68}{}%
+\contentsline {subsection}{54\ \ ETH-Schranke zeigen für $k$-IndependentSet}{69}{}%
+\contentsline {subsection}{55\ \ ETH-Schranke zeigen für 3-DM}{70}{}%
+\contentsline {subsection}{56\ \ ETH-Schranke zeigen für 3-ExactCover}{71}{}%
+\contentsline {subsection}{57\ \ ETH-Schranke zeigen für SubsetSum}{72}{}%
+\contentsline {subsection}{58\ \ ETH-Schranke zeigen für $k$-Color}{73}{}%
+\contentsline {subsection}{59\ \ ETH-Schranke zeigen für CoverClique}{74}{}%
+\contentsline {subsection}{60\ \ ETH-Schranke zeigen für $\Delta $Cover}{75}{}%
+\contentsline {subsection}{61\ \ ETH-Schranke zeigen für NAE-4-SAT}{76}{}%
+\contentsline {subsection}{62\ \ ETH-Schranke zeigen für SetSplitting}{77}{}%
+\contentsline {subsection}{63\ \ ETH-Schranke zeigen für DominatingSet}{78}{}%
+\contentsline {subsection}{64\ \ ETH-Schranke zeigen für GridTiling}{79}{}%
+\contentsline {subsection}{65\ \ ETH-Schranke zeigen für SetCover}{80}{}%
+\contentsline {subsection}{66\ \ ETH-Schranke zeigen für Scheduling $2\,|\,\mathrm {prec}, p_i \in \{1,2\}\,|\,C_{\max }$}{81}{}%
+\contentsline {subsection}{67\ \ ETH-Schranke zeigen für ILP-Feasibility}{82}{}%

lernmaterial/aufgabenkatalog.aux

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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}%
+\contentsline {subsection}{A61\ \ Fragen über Fragen I}{92}{section*.62}%
+\contentsline {subsection}{A62\ \ Fragen über Fragen II}{93}{section*.63}%
+\contentsline {subsection}{A63\ \ Wenn die ETH fehlschlägt, gilt $P = NP$?}{94}{section*.64}%
+\contentsline {section}{\numberline {7}Turingmaschinen}{95}{section.7}%
+\contentsline {subsection}{A64\ \ TM für Palindrome}{95}{section*.65}%
+\contentsline {subsection}{A65\ \ TM für $L = \{0^{2^n}\}$ + RAM-Vergleich}{96}{section*.66}%

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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}

lernmaterial/aufgabentypen.toc

@@ -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}%
+\contentsline {subsection}{\numberline {4.1}Woran du diesen Typ erkennst}{10}{subsection.4.1}%
+\contentsline {subsection}{\numberline {4.2}Das Rezept}{10}{subsection.4.2}%
+\contentsline {subsection}{\numberline {4.3}Musteraufgabe}{11}{subsection.4.3}%
+\contentsline {subsection}{\numberline {4.4}Bewertung und häufige Fehler}{11}{subsection.4.4}%
+\contentsline {section}{\numberline {5}Approximationsalgorithmus anwenden}{12}{section.5}%
+\contentsline {subsection}{\numberline {5.1}Woran du diesen Typ erkennst}{12}{subsection.5.1}%
+\contentsline {subsection}{\numberline {5.2}Das Rezept}{12}{subsection.5.2}%
+\contentsline {subsection}{\numberline {5.3}Musteraufgabe}{13}{subsection.5.3}%
+\contentsline {subsection}{\numberline {5.4}Zweite Variante: Christofides mit allen Zwischenschritten}{14}{subsection.5.4}%
+\contentsline {subsection}{\numberline {5.5}Bewertung und häufige Fehler}{14}{subsection.5.5}%
+\contentsline {section}{\numberline {6}Güte beweisen und Worst-Case-Instanz bauen}{14}{section.6}%
+\contentsline {subsection}{\numberline {6.1}Das Rezept}{15}{subsection.6.1}%
+\contentsline {subsection}{\numberline {6.2}Musteraufgabe}{15}{subsection.6.2}%
+\contentsline {subsection}{\numberline {6.3}Zweite Variante: Approximate Subset Sum (Klausur SS24-H)}{16}{subsection.6.3}%
+\contentsline {subsection}{\numberline {6.4}Bewertung und häufige Fehler}{17}{subsection.6.4}%
+\contentsline {section}{\numberline {7}Turingmaschine entwerfen}{17}{section.7}%
+\contentsline {subsection}{\numberline {7.1}Das Rezept}{17}{subsection.7.1}%
+\contentsline {subsection}{\numberline {7.2}Musteraufgabe}{17}{subsection.7.2}%
+\contentsline {subsection}{\numberline {7.3}Zweite Variante: $L = \{0^{2^n}\}$ (Präsenz 10.3)}{18}{subsection.7.3}%
+\contentsline {subsection}{\numberline {7.4}Bewertung und häufige Fehler}{19}{subsection.7.4}%

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lernmaterial/cheatsheet.tex

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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}

lernmaterial/fundament.tex

@@ -0,0 +1,858 @@
+\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}

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+\contentsline {subsection}{\numberline {1.1}Freunde auf einer Party}{3}{subsection.1.1}%
+\contentsline {subsection}{\numberline {1.2}Der naive Weg -- und warum er explodiert}{3}{subsection.1.2}%
+\contentsline {subsection}{\numberline {1.3}Die Frage, um die es geht}{4}{subsection.1.3}%
+\contentsline {section}{\numberline {2}Was heißt eigentlich \quotedblbase schnell\textquotedblleft ?}{5}{section.2}%
+\contentsline {subsection}{\numberline {2.1}Laufzeit hängt von der Eingabegröße ab}{5}{subsection.2.1}%
+\contentsline {subsection}{\numberline {2.2}Polynomiell gegen exponentiell -- die entscheidende Grenze}{5}{subsection.2.2}%
+\contentsline {subsection}{\numberline {2.3}Warum wir Probleme als Ja/Nein-Fragen schreiben}{6}{subsection.2.3}%
+\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}Die zweite Brille: Raten}{8}{subsection.3.3}%
+\contentsline {subsection}{\numberline {3.4}P steckt in NP -- und die Millionenfrage}{9}{subsection.3.4}%
+\contentsline {section}{\numberline {4}\quotedblbase Mindestens so schwer wie\textquotedblleft {} -- Reduktionen}{10}{section.4}%
+\contentsline {subsection}{\numberline {4.1}Ein Problem mit einem anderen lösen}{10}{subsection.4.1}%
+\contentsline {subsection}{\numberline {4.2}Die Richtung -- der Fehler, der alle Punkte kostet}{11}{subsection.4.2}%
+\contentsline {subsection}{\numberline {4.3}Reduktionen lassen sich verketten}{11}{subsection.4.3}%
+\contentsline {section}{\numberline {5}Die härtesten Probleme}{12}{section.5}%
+\contentsline {subsection}{\numberline {5.1}NP-schwer und NP-vollständig}{12}{subsection.5.1}%
+\contentsline {subsection}{\numberline {5.2}Warum ein einziger Algorithmus alles verändern würde}{13}{subsection.5.2}%
+\contentsline {subsection}{\numberline {5.3}Das erste NP-vollständige Problem: Cook--Levin}{13}{subsection.5.3}%
+\contentsline {subsection}{\numberline {5.4}Der Motor: das Vererbungskorollar}{13}{subsection.5.4}%
+\contentsline {section}{\numberline {6}Der Problem-Zoo}{14}{section.6}%
+\contentsline {subsection}{\numberline {6.1}Die Landkarte: eine Kette ab SAT}{14}{subsection.6.1}%
+\contentsline {subsection}{\numberline {6.2}Die Logik-Probleme}{15}{subsection.6.2}%
+\contentsline {subsection}{\numberline {6.3}Die Graph-Probleme und ihr Dreieck}{15}{subsection.6.3}%
+\contentsline {subsection}{\numberline {6.4}Eine Reduktion ganz durchgerechnet: $\textsc {SAT} \leq _{p}\textsc {$k$-Clique}$}{16}{subsection.6.4}%
+\contentsline {subsection}{\numberline {6.5}Die Zahl- und Mengen-Probleme}{17}{subsection.6.5}%
+\contentsline {subsection}{\numberline {6.6}Zwei kurze, elegante Reduktionen}{18}{subsection.6.6}%
+\contentsline {section}{\numberline {7}Wie schwer \emph {genau}? Die ETH}{18}{section.7}%
+\contentsline {subsection}{\numberline {7.1}Die Lücke, die NP-Vollständigkeit lässt}{19}{subsection.7.1}%
+\contentsline {subsection}{\numberline {7.2}Die Hypothese und warum man sie braucht}{19}{subsection.7.2}%
+\contentsline {section}{\numberline {8}Damit leben: gute Näherungen}{20}{section.8}%
+\contentsline {subsection}{\numberline {8.1}Was heißt \quotedblbase gute Näherung\textquotedblleft ?}{20}{subsection.8.1}%
+\contentsline {subsection}{\numberline {8.2}TSP: von \quotedblbase unmöglich\textquotedblleft {} zu Güte $1{,}5$}{21}{subsection.8.2}%
+\contentsline {subsection}{\numberline {8.3}Knapsack: warum Gier allein scheitert}{23}{subsection.8.3}%
+\contentsline {subsection}{\numberline {8.4}Scheduling: einfache Regeln, scharfe Schranken}{23}{subsection.8.4}%
+\contentsline {subsection}{\numberline {8.5}MAX-3-SAT: Güte 2 fast geschenkt}{24}{subsection.8.5}%
+\contentsline {section}{\numberline {9}Jenseits von NP: das Halteproblem}{25}{section.9}%
+\contentsline {section}{\numberline {10}Das große Ganze}{25}{section.10}%
+\contentsline {subsection}{\numberline {10.1}Die Reise in einem Bild}{25}{subsection.10.1}%
+\contentsline {subsection}{\numberline {10.2}Verständnis-Check: wahr oder falsch?}{25}{subsection.10.2}%
+\contentsline {subsection}{\numberline {10.3}Wie es weitergeht}{26}{subsection.10.3}%

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+\contentsline {subsection}{\numberline {1.1}Cliquen in einem Graphen}{3}{subsection.1.1}%
+\contentsline {subsection}{\numberline {1.2}Warum der naive Weg explodiert}{3}{subsection.1.2}%
+\contentsline {subsection}{\numberline {1.3}Die eigentliche Frage}{4}{subsection.1.3}%
+\contentsline {section}{\numberline {2}Was heißt \quotedblbase schnell\textquotedblleft ?}{4}{section.2}%
+\contentsline {subsection}{\numberline {2.1}Laufzeit misst man in der Eingabegröße}{5}{subsection.2.1}%
+\contentsline {subsection}{\numberline {2.2}Probleme als Ja/Nein-Fragen}{5}{subsection.2.2}%
+\contentsline {section}{\numberline {3}Prüfen ist leichter als Lösen}{6}{section.3}%
+\contentsline {subsection}{\numberline {3.1}Die Sudoku-Beobachtung}{6}{subsection.3.1}%
+\contentsline {subsection}{\numberline {3.2}Der NP-Dreischritt -- so zeigt man \quotedblbase $\in \mathsf {NP}$\textquotedblleft }{6}{subsection.3.2}%
+\contentsline {subsection}{\numberline {3.3}P steckt in NP -- und die Millionenfrage}{7}{subsection.3.3}%
+\contentsline {section}{\numberline {4}\quotedblbase Mindestens so schwer wie\textquotedblleft {} -- Reduktionen}{7}{section.4}%
+\contentsline {subsection}{\numberline {4.1}Ein Problem mit einem anderen lösen}{8}{subsection.4.1}%
+\contentsline {subsection}{\numberline {4.2}Die Richtung -- der teuerste Fehler}{8}{subsection.4.2}%
+\contentsline {section}{\numberline {5}Die härtesten Probleme: NP-vollständig}{9}{section.5}%
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+\contentsline {subsection}{\numberline {5.2}Die Landkarte der Schwere}{10}{subsection.5.2}%
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+\contentsline {section}{\numberline {7}Wie schwer \emph {genau}? Die ETH}{16}{section.7}%
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+\contentsline {section}{\numberline {9}Jenseits von NP: das Halteproblem}{21}{section.9}%
+\contentsline {section}{\numberline {10}Das große Ganze}{21}{section.10}%
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+\@writefile{toc}{\contentsline {subsection}{\numberline {11.1}Die Reise auf einen Blick}{36}{subsection.11.1}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.2}Wahr oder falsch? (Die Begründung zählt)}{36}{subsection.11.2}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\numberline {A}Lösungen zu den Abruf-Fragen}{36}{appendix.A}\protected@file@percent }
+\newlabel{sec:loesungen}{{A}{36}{Lösungen zu den Abruf-Fragen}{appendix.A}{}}
+\gdef \@abspage@last{38}