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hausaufgaben/meine_loeasung/compile.log
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Ok [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139]
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[95] [127] [32])
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[59] [61] [96] [189] [17] [45] [16] [21] [22])
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(/usr/share/texlive/texmf-dist/fonts/source/jknappen/ec/exsign.mf
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(/usr/share/texlive/texmf-dist/fonts/source/jknappen/ec/exrlig.mf
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Ok [25] [26] [28] [27] [29] [30] [31])
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(/usr/share/texlive/texmf-dist/fonts/source/jknappen/ec/exromp.mf
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Ok [38] [63] [190])
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Ok [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78]
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[79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90])
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(/usr/share/texlive/texmf-dist/fonts/source/jknappen/ec/exrllett.mf
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Ok [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108]
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[109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120]
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[121] [122])
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(/usr/share/texlive/texmf-dist/fonts/source/jknappen/ec/exrdigit.mf
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Ok [48] [49] [50] [51] [52] [53] [54] [55] [56] [57])
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(/usr/share/texlive/texmf-dist/fonts/source/jknappen/ec/exrligtb.mf
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5
hausaufgaben/meine_loeasung/hausaufgabe_10_loesung.aux
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hausaufgaben/meine_loeasung/hausaufgabe_10_loesung.aux
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hausaufgaben/meine_loeasung/hausaufgabe_10_loesung.pdf
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hausaufgaben/meine_loeasung/hausaufgabe_10_loesung.tex
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hausaufgaben/meine_loeasung/hausaufgabe_10_loesung.tex
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\documentclass[12pt]{article}
|
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|
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage[ngerman,provide=*]{babel}
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\usepackage{amsmath}
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\usepackage{amssymb}
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||||
\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}
|
||||
5
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[3] (./hausaufgabe_11_loesung.aux)
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***********
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hausaufgaben/meine_loeasung/hausaufgabe_11_loesung.pdf
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\documentclass[12pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage[ngerman,provide=*]{babel}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage[a4paper,margin=2.5cm]{geometry}
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\title{Hausaufgabe 11}
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\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
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\date{29.\ Juni 2026}
|
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\begin{document}
|
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\maketitle
|
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\section*{Hausaufgabe 11.1}
|
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\subsection*{Teil 1: $\Delta$-Cover ist in NP}
|
||||
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||||
Eingabe: Graph $G = (V, E)$, Zahl $k$ und Zertifikat $C_\Delta \subseteq V$. \\
|
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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 $\{a,b\}, \{b,c\}, \{a,c\} \in E$).
|
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\item Lehne ab, falls ein solches Dreieck kein $a$, $b$ oder $c$ in
|
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$C_\Delta$ hat.
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\item Sonst akzeptiere.
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||||
\end{itemize}
|
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Existiert eine Dreiecksüberdeckung $C_\Delta$ mit $|C_\Delta| \le k$, dann gibt
|
||||
es ein Zertifikat dafür. Dieses wird vom Verifizierer akzeptiert und sonst
|
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abgelehnt. \\
|
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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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||||
5
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\title{Hausaufgabe 12}
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\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
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\date{6.\ Juli 2026}
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\section*{Hausaufgabe 12.1}
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\subsection*{Teil (a): Der Algorithmus hat Güte 2}
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Zu zeigen ist, dass der Algorithmus die Güte 2 hat, also die Lösung vom
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Algorithmus mindestens halb so gut ist wie die optimale Lösung. Also
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$\nu(A(\varphi)) \ge \frac{1}{2}\nu(OPT(\varphi))$ für alle Eingaben $\varphi$. \\
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||||
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||||
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}
|
||||
5
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105
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\documentclass[12pt]{article}
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\title{Hausaufgabe 9}
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\author{Marek Lenczewski \\ Matrikelnummer: 1025252}
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\date{15.\ Juni 2026}
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\begin{document}
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\maketitle
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\section*{Hausaufgabe 9.1}
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\subsection*{Teil 1}
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Ein minimales Vertex Cover ist $C = \{2, 7, 5, 4, 1\}$.
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\subsection*{Teil 2}
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Der Algorithmus arbeitet wie folgt:
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\begin{itemize}
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||||
\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}
|
||||
Reference in New Issue
Block a user