From 833a8b5c5d21143da99be847c805f1f8a99d1c3f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 8 Dec 2025 10:14:29 +0100 Subject: [PATCH] [TI] Compact: Start NP-completeness --- semester3/ti-compact/parts/05_complexity.tex | 24 +++++++++++++++++++ semester3/ti-compact/ti-compact.pdf | Bin 373508 -> 378427 bytes 2 files changed, 24 insertions(+) diff --git a/semester3/ti-compact/parts/05_complexity.tex b/semester3/ti-compact/parts/05_complexity.tex index 41a0cb9..f4b86c7 100644 --- a/semester3/ti-compact/parts/05_complexity.tex +++ b/semester3/ti-compact/parts/05_complexity.tex @@ -157,3 +157,27 @@ If we combine some of the above results, we get: \inlinetheorem $VP = NP$ \subsection{NP-Completeness} +\fancydef{Polynomial Reduction} $L_1 \leq_p L_2$, if there exists polynomial TM $A$ with $x \in L_1 \Leftrightarrow A(x) \in L_2$. +$A$ is called a polynomial reduction of $L_1$ into $L_2$ + +\begin{definition}[]{$NP$-Hard and $NP$-Complete} + A language $L$ is called $NP$-hard, if for all $L' \in NP$, we have $L' \leq_p L$ + + A language $L$ is called $NP$-complete, if it is $NP$-hard and and $L \in NP$ +\end{definition} + +\inlinelemma If $L \in P$ and $L$ is $NP$-hard, then $P = NP$ + +\fancydef{Cook} $SAT$ is $NP$-complete + +\inlinelemma If $L_1 \leq_p L_2$ and $L_1$ is $NP$-hard, then $L_2$ is $NP$-hard + +A few languages commonly used to show $NP$-completeness: +\begin{itemize} + \item $SAT = \{ \Phi \divides \Phi \text{ is a satisfiable formula in CNF} \}$ + \item $\text{CLIQUE} = \{ (G, k) \divides G \text{ is an undirected graph that contains a $k$-clique } \}$ + \item $VC = \{ (G, k) \divides G \text{ is an undirected graph with a vertex cover of size $\leq k$ } \}$ + \item $3SAT = \{ \Phi \divides \Phi \text{ is a satisfiable formula in CNF with all clauses containing \textit{at most} three literals} \}$ +\end{itemize} +where a clique is \TODO Add. +and a vertex cover is any set $U \subseteq V$ where all edges $\{ u, v \} \in E$ have at least one endpoint 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