From 68ee2ec327e1a3a5b9227f59b9fbfb79c7ad1dbc Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 26 Nov 2025 16:44:51 +0100 Subject: [PATCH] [TI] Almost catch up --- .../02_non-deterministic-complexity.tex | 2 +- .../05_complexity/04_np-completeness.tex | 32 +++++++++++++++++- semester3/ti/ti-summary.pdf | Bin 1011249 -> 1014744 bytes 3 files changed, 32 insertions(+), 2 deletions(-) diff --git a/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex b/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex index dd9ba30..a1b488b 100644 --- a/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex +++ b/semester3/ti/parts/05_complexity/02_non-deterministic-complexity.tex @@ -40,7 +40,7 @@ \drmvspace \inlinecorollary $\text{NP} \subseteq \text{PSPACE}$ -\inlineremark Für jede platzkonstruierbare Funktion $s$ mit $s(n) \geq \log_2(n)$ gilt +\inlinetheorem Für jede platzkonstruierbare Funktion $s$ mit $s(n) \geq \log_2(n)$ gilt \rmvspace \begin{align*} \text{NSPACE}(s(n)) \subseteq \bigcup_{c \in \N} \text{TIME}(c^{s(n)}) diff --git a/semester3/ti/parts/05_complexity/04_np-completeness.tex b/semester3/ti/parts/05_complexity/04_np-completeness.tex index 6c6cb5b..f24b44d 100644 --- a/semester3/ti/parts/05_complexity/04_np-completeness.tex +++ b/semester3/ti/parts/05_complexity/04_np-completeness.tex @@ -32,13 +32,43 @@ Wieder bedeutet $L_1 \leq_p L_2$, dass $L_2$ mindestens so schwer ist wie $L_1$ \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. +Jedoch sind diese Beweise sehr gross und deshalb nicht prüfungsrelevant. \inlinelemma Falls $L_1 \leq_p L_2$ und $L_1$ ist $NP$-Schwer, so ist auch $L_2$ $NP$-Schwer -Betrachten wir folgende +Betrachten wir folgende Sprachen: +\begin{align*} + SAT & = \{ \Phi \divides \Phi \text{ ist eine erfüllbare Formel in CNF} \} \\ + CLIQUE & = \{ (G, k) \divides G \text{ ist ein ungerichteter Graph, der eine $k$-clique enthält} \} \\ + VC & = \{ (G, k) \divides G \text{ ist ein ungerichteter Graph mit einer Kontenüberdeckung der Mächtigkeit höchstens } k \} +\end{align*} +Wir erinnern uns daran, dass eine Kontenüberdeckung eines Graphen $G = (V, E)$ jede Menge von Konten $U \subseteq V$ ist, +so dass jede Kante aus $E$ mindestens einen Endpunkt in $U$ hat. + +\inlinelemma $SAT \leq_p CLIQUE$ + +\inlinelemma $CLIQUE \leq_p VC$ + +\inlinelemma $SAT \leq_p 3SAT$, wobei wir beim $3SAT$-Problem bestimmen wollen, ob eine Formel in $3CNF$ (CNF, aber alle Klauseln enthalten höchstens $3$ Variabeln) erfüllbar ist. + + +\begin{definition}[]{$NPO$} + $NPO$ ist die Klasse der Optimierungsprobleme, mit $U = (\Sigma_I, \Sigma_O, L, \cM, \text{cost}, \text{goal}) \in NPO$, falls folgende Bedingungen erfüllt sind: + \begin{enumerate}[label=\textit{(\roman*)}] + \item $L \in P$ + \item Es existiert ein Polynom $p_U$, so dass + \begin{enumerate}[label=(\alph*)] + \item Für jedes $x \in L$ und jedes $y \in \cM(x)$ + \item es existiert ein polynomieller Algorithmus $A$, der für jedes $y \in \word_O$ und jedes $x \in L$ mit $|y| \leq p_U(|x|)$ entscheidet, + ob $y \in \cM(x)$ oder nicht + \end{enumerate} + \item Die Funktion $\text{cost}$ kann man in polynomieller Zeit berechnen. + \end{enumerate} +\end{definition} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index a03192e5844ed665fc3c893bfc5af4726b053b2c..21f3b1a727ed77701bc0322b0d374767cff9f744 100644 GIT binary patch delta 81019 zcmV)LK)JuM+&$R$J%EG(gaU*Egam{Iga(8Mgb1_=lB@$UHZqq{0SGC79ocdlH}YLy zG5bc+$7cyqYY} zj=3GY9v=^dkc02$uPzsVv(aku_W0M+?<)yC86fpNz8}peOKc_v|Grv%zy?R(o*w;v zge_>r(lCrL>#DQa(TiVcbq>$IuP71DR)6}IGia#^)dKFP)$^lgM>Hp0%8D`Kl;Sz< zN)yFgphOdSJYhZlD4_t)z@yc2Hd-#nak8DMYRL#S5~uom1+#cu`KcWF*(#VWg2Vp- 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