From 30f9d722df8a02471efa09b34bcfe53cde8a0359 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 26 Nov 2025 15:28:24 +0100 Subject: [PATCH] [TI] Finish class NP, start NP completeness --- .../ti/parts/05_complexity/03_class-np.tex | 4 ++ .../05_complexity/04_np-completeness.tex | 42 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 1006300 -> 1011249 bytes 3 files changed, 46 insertions(+) diff --git a/semester3/ti/parts/05_complexity/03_class-np.tex b/semester3/ti/parts/05_complexity/03_class-np.tex index 0ee3acf..64f08a4 100644 --- a/semester3/ti/parts/05_complexity/03_class-np.tex +++ b/semester3/ti/parts/05_complexity/03_class-np.tex @@ -29,3 +29,7 @@ Für nichtdeterministische Berechnungen nennen wir $\alpha_1, \ldots, \alpha_n$ \end{align*} \end{enumerate} \end{definition} + +\inlinetheorem $VP = NP$ + +Der Beweis für obiges Resultat ist auf Seiten 193 - 194 im Buch (= 205 - 206 im PDf) zu finden diff --git a/semester3/ti/parts/05_complexity/04_np-completeness.tex b/semester3/ti/parts/05_complexity/04_np-completeness.tex index 2a61bde..6c6cb5b 100644 --- a/semester3/ti/parts/05_complexity/04_np-completeness.tex +++ b/semester3/ti/parts/05_complexity/04_np-completeness.tex @@ -1,2 +1,44 @@ \newpage \subsection{NP-Vollständigkeit} +Es sind mittlerweile über 3000 Probleme bekannt, für welche wir keinen Algorithmus kennen, der in polynomieller Zeit läuft. +Es ist aber bis jetzt niemandem gelungen, eine höhere untere Schranke für alle zu beweisen, als $\tcl{n}$. + +Wie bereits bei der Berechenbarkeit benutzen wir eine Reduktion. +Falls jedes Problem aus $NP$ effizient auf ein Problem $L \in NP$ reduzierbar ist, so ist $L$ schwer. + +\begin{definition}[]{Polynomielle Reduktion} + $L_1 \subseteq \word_1$ ist \bi{polynomiell reduzierbar auf} $L_2 \subseteq \word_2$, geschrieben $L_1 \leq_p L_2$, + falls eine polynomielle TM $A$ existiert, die für jedes Wort $x \in \word_1$ ein Word $A(x) \in \word_2$ berechnet, so dass + \rmvspace + \begin{align*} + x \in L_1 \Longleftrightarrow A(x) \in L_2 + \end{align*} + + \drmvspace + $A$ wird eine polynomielle Reduktion von $L_1$ auf $L_2$ genannt. +\end{definition} +Wieder bedeutet $L_1 \leq_p L_2$, dass $L_2$ mindestens so schwer ist wie $L_1$ + +\begin{definition}[]{$NP$-Schwer} + Eine Sprache $L$ ist \bi{$NP$-Schwer}, falls für alle $L' \in NP$ gilt $L' \leq_p L$. + + Eine Sprache $L$ ist \bi{$NP$-Vollständig}, falls + \drmvspace + \begin{multicols}{2} + \begin{enumerate}[label=\textit{(\roman*)}] + \item $L \in NP$ + \item $L$ $NP$-Schwer ist. + \end{enumerate} + \end{multicols} +\end{definition} + +\inlinelemma Falls $L \in P$ und $L$ ist $NP$-schwer, dann gilt $P = NP$ + +\fancytheorem{Cook} $SAT$ ist $NP$-Vollständig + +Der Beweis hierfür liefert eine grobe Struktur für weitere Beweise dieser Art und ist auf Seiten 199 - 205 im Buch (= Seiten 211 - 217 im PDF) zu finden. + + +\inlinelemma Falls $L_1 \leq_p L_2$ und $L_1$ ist $NP$-Schwer, so ist auch $L_2$ $NP$-Schwer + +Betrachten wir 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