\contentsline {section}{\numberline {1}NP-Zugehörigkeit}{2}{section.1}%
\contentsline {subsection}{A1\ \ Longest Path $\in $ NP (Dreischritt)}{2}{section*.2}%
\contentsline {subsection}{A2\ \ Dominating Set $\in $ NP (Dreischritt)}{4}{section*.3}%
\contentsline {section}{\numberline {2}Vorlesungsbeweise}{6}{section.2}%
\contentsline {subsection}{A3\ \ $P \subseteq NP$}{6}{section*.4}%
\contentsline {subsection}{A4\ \ NDTM-Definition $\subseteq $ Verifizierer-Definition}{7}{section*.5}%
\contentsline {subsection}{A5\ \ Verifizierer-Definition $\subseteq $ NDTM-Definition}{9}{section*.6}%
\contentsline {subsection}{A6\ \ Transitivität von Polynomialzeitreduktionen}{10}{section*.7}%
\contentsline {subsection}{A7\ \ Vererbung der NP-Vollständigkeit}{11}{section*.8}%
\contentsline {subsection}{A8\ \ $L_0$ NP-vollständig und $L_0 \in P$ $\Rightarrow $ $P = NP$}{12}{section*.9}%
\contentsline {subsection}{A9\ \ SAT $\leq _{p}$ 3-SAT}{13}{section*.10}%
\contentsline {subsection}{A10\ \ $k$-Clique ist NP-vollständig}{15}{section*.11}%
\contentsline {subsection}{A11\ \ SubsetSum $\leq _{p}$ Knapsack}{17}{section*.12}%
\contentsline {subsection}{A12\ \ SubsetSum $\leq _{p}$ Partition}{18}{section*.13}%
\contentsline {subsection}{A13\ \ ListScheduling hat Güte $2 - \frac {1}{m}$}{20}{section*.14}%
\contentsline {subsection}{A14\ \ ModifiedGreedy hat Güte 2}{22}{section*.15}%
\contentsline {section}{\numberline {3}Reduktionen: NP-Schwere und NP-Vollständigkeit}{23}{section.3}%
\contentsline {subsection}{A15\ \ VertexCover ist NP-vollständig}{23}{section*.16}%
\contentsline {subsection}{A16\ \ $k$-CliqueUniversal ist NP-vollständig}{25}{section*.17}%
\contentsline {subsection}{A17\ \ Clique-Nomember ist NP-vollständig}{27}{section*.18}%
\contentsline {subsection}{A18\ \ CliqueAndIndependentSet ist NP-schwer}{29}{section*.19}%
\contentsline {subsection}{A19\ \ $k$-CLIQUE-DEG-3 ist NP-vollständig}{31}{section*.20}%
\contentsline {subsection}{A20\ \ FeedbackVertexSet ist NP-vollständig}{34}{section*.21}%
\contentsline {subsection}{A21\ \ $\Delta $-Cover ist NP-vollständig}{36}{section*.22}%
\contentsline {subsection}{A22\ \ 3-COLOR mit Minimalgrad 3 ist NP-vollständig}{38}{section*.23}%
\contentsline {subsection}{A23\ \ $k$-COLOR-PRECOLORING ist NP-vollständig}{40}{section*.24}%
\contentsline {subsection}{A24\ \ HamiltonianPath $\leq _{p}$ HamiltonianCycle}{42}{section*.25}%
\contentsline {subsection}{A25\ \ HamiltonianCycle $\leq _{p}$ HamiltonianPath}{44}{section*.26}%
\contentsline {subsection}{A26\ \ Hitchhiker's-HamiltonianCycle ist NP-schwer}{46}{section*.27}%
\contentsline {subsection}{A27\ \ SubsetSumCardinality ist NP-vollständig}{48}{section*.28}%
\contentsline {subsection}{A28\ \ $(a_1{=}1)$-SubsetSum ist NP-schwer}{49}{section*.29}%
\contentsline {subsection}{A29\ \ AtMostTwoPerSize-SubsetSum ist NP-schwer}{50}{section*.30}%
\contentsline {subsection}{A30\ \ SubsetSum mit Teilbarkeit ist NP-vollständig}{52}{section*.31}%
\contentsline {subsection}{A31\ \ SubsetSum ohne Zweierpotenzen ist NP-vollständig}{53}{section*.32}%
\contentsline {subsection}{A32\ \ TSP ist nicht approximierbar}{54}{section*.33}%
\contentsline {subsection}{A33\ \ HALT$_{\text {TM}}$ ist NP-schwer}{55}{section*.34}%
\contentsline {section}{\numberline {4}ETH-Schranken}{56}{section.4}%
\contentsline {subsection}{A34\ \ Lower Bounds: VertexCover}{56}{section*.35}%
\contentsline {subsection}{A35\ \ Lower Bounds: HittingSet}{58}{section*.36}%
\contentsline {subsection}{A36\ \ Lower Bounds entlang der Reduktionskette}{60}{section*.37}%
\contentsline {subsection}{A37\ \ Lower Bounds: $k$-Color}{62}{section*.38}%
\contentsline {subsection}{A38\ \ Lower Bounds: $2\,|\,\mathrm {prec}, p_i \in \{1,2\}\,|\,C_{\max }$}{64}{section*.39}%
\contentsline {subsection}{A39\ \ Lower Bounds: DominatingSet}{65}{section*.40}%
\contentsline {subsection}{A40\ \ Lower Bounds: GridTiling}{66}{section*.41}%
\contentsline {subsection}{A41\ \ Lower Bounds: SubsetSum (strenge Reduktion)}{67}{section*.42}%
\contentsline {subsection}{A42\ \ Lower Bounds: SetCover}{68}{section*.43}%
\contentsline {subsection}{A43\ \ Lower Bounds: ILP-Feasibility}{69}{section*.44}%
\contentsline {subsection}{A44\ \ Lower Bounds: NAE-4-SAT}{70}{section*.45}%
\contentsline {subsection}{A45\ \ Lower Bounds: SetSplitting}{71}{section*.46}%
\contentsline {subsection}{A46\ \ Lower Bounds: CoverClique}{73}{section*.47}%
\contentsline {subsection}{A47\ \ Lower Bounds: $\Delta $Cover}{74}{section*.48}%
\contentsline {section}{\numberline {5}Approximation: anwenden, Güte beweisen, Worst Case}{75}{section.5}%
\contentsline {subsection}{A48\ \ Knapsack: Greedy und ModifiedGreedy}{75}{section*.49}%
\contentsline {subsection}{A49\ \ LPT anwenden}{77}{section*.50}%
\contentsline {subsection}{A50\ \ ListScheduling: scharfe Worst-Case-Instanzen}{78}{section*.51}%
\contentsline {subsection}{A51\ \ 2ApproxVC hat Güte 2}{79}{section*.52}%
\contentsline {subsection}{A52\ \ MAX-3-SAT: Güte 2 + scharfe Instanz}{80}{section*.53}%
\contentsline {subsection}{A53\ \ ApproximateSubsetSum: Worst Case + Güte 2}{81}{section*.54}%
\contentsline {subsection}{A54\ \ RoundRobin: Güte 2 + Worst Case}{82}{section*.55}%
\contentsline {subsection}{A55\ \ Min-Edge-Cover: Güte 2 + Worst Case im Kreis}{83}{section*.56}%
\contentsline {subsection}{A56\ \ Christofides-Rate $3/2$ ist scharf}{84}{section*.57}%
\contentsline {subsection}{A57\ \ Christofides anwenden ($K_4$)}{85}{section*.58}%
\contentsline {subsection}{A58\ \ $\Delta $TSP1 anwenden + Verständnisfragen}{86}{section*.59}%
\contentsline {subsection}{A59\ \ Parallel-Task-Scheduling}{88}{section*.60}%
\contentsline {subsection}{A60\ \ Strip Packing: NFDH}{90}{section*.61}%
\contentsline {section}{\numberline {6}Wahr oder falsch?}{92}{section.6}%
\contentsline {subsection}{A61\ \ Fragen über Fragen I}{92}{section*.62}%
\contentsline {subsection}{A62\ \ Fragen über Fragen II}{93}{section*.63}%
\contentsline {subsection}{A63\ \ Wenn die ETH fehlschlägt, gilt $P = NP$?}{94}{section*.64}%
\contentsline {section}{\numberline {7}Turingmaschinen}{95}{section.7}%
\contentsline {subsection}{A64\ \ TM für Palindrome}{95}{section*.65}%
\contentsline {subsection}{A65\ \ TM für $L = \{0^{2^n}\}$ + RAM-Vergleich}{96}{section*.66}%