From b6ddb83224ab77478b4e93424281dc4d1fbe7491 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 2 Dec 2025 10:16:14 +0100 Subject: [PATCH] [TI] Catch up --- .../05_complexity/04_np-completeness.tex | 50 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 1014744 -> 1018893 bytes 2 files changed, 50 insertions(+) diff --git a/semester3/ti/parts/05_complexity/04_np-completeness.tex b/semester3/ti/parts/05_complexity/04_np-completeness.tex index f24b44d..28fa94c 100644 --- a/semester3/ti/parts/05_complexity/04_np-completeness.tex +++ b/semester3/ti/parts/05_complexity/04_np-completeness.tex @@ -72,3 +72,53 @@ so dass jede Kante aus $E$ mindestens einen Endpunkt in $U$ hat. \item Die Funktion $\text{cost}$ kann man in polynomieller Zeit berechnen. \end{enumerate} \end{definition} + +\newpage +Ein Optimierungsproblem $U$ ist also in $NPO$, falls +\rmvspace +\begin{enumerate}[noitemsep] + \item man effizient überprüfen kann, ob ein gegebenes Wort ein Problemfall von $U$ ist + \item die Grösse der Lösungen polynomiell in der Grösse des Problemfalls (Eingabe) und in polynomieller Zeit verifizert werden kann, + ob $y$ eine zulässige Lösung für einen gegebenen Problemfall ist + \item man die Kosten der zulässigen Lösung effizient berechnen kann +\end{enumerate} + +$\text{MAX-SAT}$ liegt in $NPO$ + +\begin{definition}[]{PO} + $PO$ ist die Klasse von Optimierungsproblemen $U = (\Sigma_I, \Sigma_O, L, \cM, \text{cost}, \text{goal})$, so dass + \begin{enumerate}[label=(\roman*)] + \item $U \in NPO$ + \item $\exists$ polynomieller Algorithmus $A$, so dass $A(x)$ für jedes $x \in L$ die optimale Lösung für $x$ ist. + \end{enumerate} +\end{definition} + +\begin{definition}[]{Schwellenwert-Sprache} + Die Schwellenwert-Sprache für $U$ (ein Optimierungsproblem aus $NPO$) ist + \rmvspace + \begin{align*} + \text{Lang}_U = \{ (x, a) \in L \times \wordbool \divides \text{Opt}_U(x) \leq \text{Nummer}(a) \} + \end{align*} + + \rmvspace + mit $\text{Opt}_U(x)$ die optimale Lösung, falls $\text{goal} = \text{Minimum}$, und + \rmvspace + \begin{align*} + \text{Lang}_U = \{ (x, a) \in L \times \wordbool \divides \text{Opt}_U(x) \leq \text{Nummer}(a) \} + \end{align*} + + \rmvspace + falls $\text{goal} = \text{Maximum}$ + + Wir sagen, dass $U$ \bi{NP-schwer} ist, falls $\text{Lang}_U$ NP-schwer ist. +\end{definition} + +\inlinelemma Falls ein Optimierungsproblem $U \in PO$, dann $\text{Lang}_U \in P$ + +\inlinetheorem Sei $U \in NPO$. Falls $U$ NP-schwer ist und $P \neq NP$, dann $U \notin PO$ + +\inlinelemma MAX-SAT ist NP-schwer. + +\inlinelemma MAX-CL (Das Problem der maximalen Clique) ist NP-schwer + +Um zu zeigen, dass solche Probleme $U$ NP-schwer sind, reicht es zu zeigen, dass $\text{Lang}_U$ NP-schwer ist, was wir mit einer $P$-Reduktion machen können. diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 21f3b1a727ed77701bc0322b0d374767cff9f744..35ed2bc6d2088031da590c68e614e119230fc3e2 100644 GIT binary patch delta 162354 zcmV)KK)S!!_dSi6KCoR00x~y~VFW0XhY1jWCqeku#*SY}?2b6iOxzr3i85)6WJ^)W z#Fzf{F1Vm9%2Lu!4sLzo>(%gCU>71nZw={VN}|LeC;_a;EXAYCD2PUo%}Xo-toRS=!GJ3};P^1ax)M_nyMch)LBB^L%#-56Jvh`!Irc z-nwNI2qr=v=sxPZ;Q{q>J2nyLno1c1!3@>RpO9Xr@6JFQ6Txk0Z1776$DR9?H#hGu zKb#{yy}9{t=pgVzQK@2#LSm@U+C>2f>$C6&m&(U1-sZjwKIU}|M<_U=bjeYFGMFPF zf;5Cwt=qDGh~8(bSy9%WBU`SvIGWciA}DLGtdABf+_pV_r8U%OOCrEZ?1BGxzXK~w zCS+@Uy;#=|C=IIHP_rN}@B2@Egfh|szvAGop;r2+&a>Vp;H$D((Rs3*6-dpq6%wnw 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