From d9745a9740bd0d150917c8ac2a68294316ac4b22 Mon Sep 17 00:00:00 2001 From: Never Gude Date: Sat, 6 Jun 2026 18:02:38 +0200 Subject: [PATCH 1/2] add autosave.ipe to gitignore --- .gitignore | 3 +++ 1 file changed, 3 insertions(+) diff --git a/.gitignore b/.gitignore index 542f714..b9d873b 100644 --- a/.gitignore +++ b/.gitignore @@ -329,3 +329,6 @@ TSWLatexianTemp* # option is specified. 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Übungsblatt} +\author{Jasper Gude \and Pia Röttgers} + +\begin{document} +\maketitle +\points[2em]{25} + +\input{aufgabe_1.tex} +\input{aufgabe_2.tex} +\input{aufgabe_3.tex} + +\end{document} diff --git a/übung_7/aufgabe_1.tex b/übung_7/aufgabe_1.tex new file mode 100644 index 0000000..eca2512 --- /dev/null +++ b/übung_7/aufgabe_1.tex @@ -0,0 +1,58 @@ +\section{LP-Runden} +\begin{tasks} + \item + Wir suchen in $U$ nach dem Knoten $v$ mit dem kleinsten $x_v$, unser $\epsilon$, + und setzen alle Knoten $u \in U$ auf $x_u - \epsilon$ und alle Knoten $w \in W$ + auf $\min(x_w~+~\epsilon, 1)$. + + Das macht unsere Beschränkung $x_u + x_w \geq 1$ nicht kaputt, da ein Nachbar $w$ + von einem Knoten $u \in U$ mindestens den Wert $\frac{1}{2} + \epsilon$ hat, + also $1$ ist oder in $W$ liegt. + + Somit hat sich $\abs{U} + \abs{W}$ um mindestens $1$ verkleinert, da + $x_v$ nun $0$ ist und somit nicht in $U \cup W$ liegt und zu $W$ keine + weiteren Knoten dazukommen können. + + Die Veränderungen der Variablen läuft in $\Oh(V)$, da im schlimmsten Fall + alle Knoten entweder in $U$ oder in $W$ liegen. + \points{4} + \item + \label{1b} + Wir teilen die Knoten $v \in V$ in drei Mengen ein. + \begin{align*} + A = &\set{v \mid x_v \leq \frac{1}{2}-\epsilon} \\ + B = &\set{v \mid v \notin A \cup C} \\ + C = &\set{v \mid x_v \geq \frac{1}{2}+\epsilon} + \end{align*} + Für alle $v \in A$ setzen wir $x_v = 0$, für alle $v \in B$ setzen wir + $x_v = \frac{1}{2}$ und für alle $v \in C$ setzen wir $x_v = 1$. + Als $\epsilon$ wählen wir das das kleinste $x_v$. + + Das dürfen wir, da wir damit die LP-Beschränkungen aufrechterhalten. + \begin{enumerate} + \item Für alle Variablen $x_v$ gilt $0 \leq x_v \leq 1$. + \item Für alle Paare $v, u$ gilt $x_v + x_u \geq 1$, da + + Fall 1: O.B.d.A gilt: Wenn $x_v \leq \frac{1}{2} - \epsilon$, dann muss $x_u \geq \frac{1}{2} + \epsilon$, wird $x_v = 0$ und $x_u = 1$. + + Fall 2: Wenn $x_v$ oder $x_u \geq \frac{1}{2} + \epsilon$, werden + $x_v$ oder $x_u = 1$. + + Fall 3: Wenn $x_v, x_u \notin A \cup C$, werden $x_v = x_u = \frac{1}{2}$. + \end{enumerate} + Die Laufzeit beläuft sich auf $\Oh(V)$ da sich das $\min_{v \in V} x_v$ in + $\Oh(V)$ Zeit finden und sich jeder Knoten in $\Oh(V)$ in eine + der drei Mengen einteilen lassen kann. + \points{3} + \item + Wir nehmen als Basis den Algorithmus aus \autoref{1b}. Um nun eine + eindeutige 2-Approximation für die Knotenüberdeckung zu bekommen, + runden wir alle $x_v = \frac{1}{2}$ auf $1$ auf. + + Dadurch erhalten wir eine Lösung die maximal doppelt so viele Knoten + enthält, wie eine optimale Lösung, da für ein Knotenpaar $v, u$ für das + $x_v = x_u = \frac{1}{2}$ mindestens ein Knoten in der Knotenüberdeckung + enthalten sein muss. Wenn wir beide Knoten nehmen, haben wir doppelt so + viele. + \points{2} +\end{tasks} diff --git a/übung_7/aufgabe_2.tex b/übung_7/aufgabe_2.tex new file mode 100644 index 0000000..3d5e438 --- /dev/null +++ b/übung_7/aufgabe_2.tex @@ -0,0 +1,75 @@ +\section{Christofides’ Algorithmus} +\begin{tasks} +\item + Siehe \autoref{fig:christofides}. + \points{4} +\item + Siehe \autoref{fig:checkmate}. Diese Tour kann nicht von \algt{Christofides} + berechnet werden, denn egal mit welcher Kante in welche Richtung gestartet + wird, nimmt \algt{Cristofides} immer eine Kante in die Tour, die nicht in + \autoref{fig:checkmate} ist. + + \begin{figure} + \centering + \includegraphics[page=10, width=0.5\textwidth]{figures.pdf} + \caption{Eine kleinere TSP-Tour, die von \algt{Cristofides} nicht berechnet werden kann.} + \label{fig:checkmate} + \end{figure} + \points{1} +\end{tasks} + +{ +\newpage +\KOMAoptions{mpinclude=false} +\recalctypearea +\begin{figure} + \centering + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=1, width=0.9\textwidth]{figures.pdf} + \caption{Der gewichtete, vollständige Graph $G$.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=2, width=0.9\textwidth]{figures.pdf} + \caption{Minimalen Spannbaum finden.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=3, width=0.9\textwidth]{figures.pdf} + \caption{Knoten $U$ mit ungeraden Graden finden.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=4, width=0.9\textwidth]{figures.pdf} + \caption{Minimales, perfektes Matching auf $G[U]$ finden.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=5, width=0.9\textwidth]{figures.pdf} + \caption{Eulertour konstruieren.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=6, width=0.9\textwidth]{figures.pdf} + \caption{Eulertour konstruieren.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=7, width=0.9\textwidth]{figures.pdf} + \caption{Eulertour konstruieren.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=8, width=0.9\textwidth]{figures.pdf} + \caption{Eulertour konstruieren.} + \end{subfigure} + \begin{subfigure}{.3\linewidth} + \centering + \includegraphics[page=9, width=0.9\textwidth]{figures.pdf} + \caption{Schon besuchte Knoten überspringen. TSP vollständig.} + \end{subfigure} + \caption{Cristofides' Algorithmus} + \label{fig:christofides} +\end{figure} +} diff --git a/übung_7/aufgabe_3.tex b/übung_7/aufgabe_3.tex new file mode 100644 index 0000000..1a11ee7 --- /dev/null +++ b/übung_7/aufgabe_3.tex @@ -0,0 +1,72 @@ +\section{Matchings in allgemeinen Graphen} +\begin{tasks} + \item + +\begin{pseudocode} +Matchings($G = \tup(V, E)$) + $M = \emptyset$ + $visited =$ |Array von $False$ der Größe| $|V|$ // Markiert gematchte Knoten + + for $e$ in $E$ do + if $\neg visited[u] \wedge \neg visited[v]$ then + $M = M \cup \{e\}$ + $visited[u] = True$ + $visited[v] = True$ + return $M$ +\end{pseudocode} + + Angenommen $M$ wäre erweiterbar (d.h. es gibt eine Kante $\{u,v\}$ mit $u,v \notin V(M)$). Dann wurden sowohl $u$ als auch $v$ während des Algorithmus beim Durchlaufen der Kante $\{u,v\}$ als frei angesehen und wäre somit der Menge $M$ hinzugefügt worden. Das ist eine Widerspruch. + + \textbf{Laufzeit:} + + Initialisierung des Arrays: $\Oh(V)$ + + Schleife: Jede Kante wird genau einmal betrachtet ($\Oh(V)$) und die Überprüfung und Markierung passieren in $\Oh(1)$. Also insgesamt $\Oh(E)$ + + Das ergibt eine Gesamtlaufzeit von $\Oh(V+E)$ + \points{3} + \item + \points{2} + + \item + Sei $M$ das vom Algorithmus berechnete maximale Matching und $M^*$ das optimale maximale Matching. + + Für jede Kante $e^*=\{u,v\}\in M^*$ gilt: mindestens einer der beiden Knoten $u$ oder $v$ muss über eine Kante aus $M$ abgedeckt werden, sonst wäre der $M$ nicht nicht-erweiterbar. + + Eine Kante aus $M$ hat genau zwei Endknoten und kann daher höchstens zwei verschiedene Kanten aus $M^*$ ,,blockieren``. Da alle Kanten in $M^*$ disjunkt sind kann man daraus folgern: + \[ + \abs{M^*} \leq 2\cdot \abs{M} \implies \abs{M} \geq \frac{1}{2} \abs{M^*} + \] + Somit ist der Algorithmus aus der Teilaufgabe a) eine 1/2 Approximation für ein optimales Matching. + + \points{3} + \item + + Zielfunktion: + \[ + \arg\min \sum_{e\in E} x_e \geq 1 + \] + Entscheidungsvariablen: für jede Kante $e\in E$: + \[ + x_e\in\{0,1\}\quad \forall e\in E + \] + Die Variable nimmt den Wert 1 an, wenn die Kante im Matching M ist und 0 falls sie das nicht ist. + + Nebenbedingungen: + \begin{enumerate} + \item + Matching: Jeder Knoten darf von maximal einer Matching-Kante berührt werden + \[ + \forall v\in V \colon \sum_{e\in \delta(v)}x_e\leq1 + \] + $\delta(v)$ ist die Menge aller Kanten welche an $v$ anliegen + \item Nicht-Erweiterbarkeit: Für jede Kante $\{u,v\}$ muss die Summe der Matching-Kanten an $u$ und $v$ mindestens 1 sein + \[ + \forall \{u,v\} \in E \colon \sum_{e \in \delta(u)} x_e + \sum_{e' \in \delta(v)} x_e' \geq 1 + \] + $\delta(v)$ ist die Menge aller Kanten welche an $v$ anliegen. Äquivalent für $\delta(u)$. + \end{enumerate} + \points{3} + \item + \points{2} +\end{tasks} diff --git a/übung_7/aufgabe_4.tex b/übung_7/aufgabe_4.tex new file mode 100644 index 0000000..4535a11 --- /dev/null +++ b/übung_7/aufgabe_4.tex @@ -0,0 +1,38 @@ +\section{Längste Wege} +\begin{tasks} + \item + Da $s, t$ in $G'$ adjazent zu jedem Knoten in $G$ ist, können wir + einen einfachen $s$-$t$-Weg der Länge $k+2$ erzeugen, indem wir + einen einfachen Weg der Länge $k$ in $G$ nehmen, $s$ an das eine Ende und $t$ an das andere Ende hängen. + + Umgekehrt kann man aus einem einfachen $s$-$t$-Weg der Länge $k$ + in $G'$ einen einfachen Weg der Länge $k-2$ in $G$ konstruieren, + indem wir $s$ und $t$ entfernen. + \points{2} + + \item + Ein Hamiltonweg ist ein Weg der alle Knoten in $G$ beinhaltet + und somit Länge $n-1$ besitzt. + + Wie wir oben gezeigt haben, kann ein $s$-$t$-Weg der Länge $n+1$ + in $G'$ leicht in einen Weg der Länge $n-1$ in $G$ umgewandelt + werden. Das heißt, dass wir einen Hamiltonweg in $G$ finden, + wenn wir einen $s$-$t$-Weg finden. + + Umgekehrt können wir einen Hamiltonweg leicht in einen $s$-$t$-Weg + umwandeln, also finden wir einen $s$-$t$-Weg wenn wir einen + Hamiltonweg finden. + + Also finden wir einen Hamiltonweg genau dann, wenn wir einen + $s$-$t$-Weg finden. + \points{1} + + \item + Da wir Hamiltonweg auf \algt{Längster $s$-$t$-Weg} reduziert + haben, muss also \algt{Längster $s$-$t$-Weg} $\NPe$-schwer sein, denn + wenn es in $\Pe$ liegen würde, könnten wir auch Hamiltonweg in + polynomieller Zeit lösen. 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b/übung_7/ngexrcs.cls @@ -0,0 +1,71 @@ +% vim: set filetype:tex + +% Identification % +\NeedsTeXFormat{LaTeX2e} +\ProvidesClass{ngexrcs}[2021/12/21 ADS Exercise class] + +% Handle options % +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{scrartcl}} +\ProcessOptions\relax + +% More declarations % +\LoadClass{scrartcl} + +\KOMAoptions{mpinclude=true} +\recalctypearea + +\RequirePackage{ngutils} + +% define fonts +\RequirePackage{fontspec} +\RequirePackage[math-style=upright]{unicode-math} +\setmainfont{TeX Gyre Pagella} +\setsansfont{TeX Gyre Heros} +\setmonofont{TeX Gyre Cursor} +\setmathfont{Euler Math} + +\setlength{\marginparwidth}{1.5\marginparwidth} +\setlength{\fboxrule}{\heavyrulewidth} + +% Use sansfont for all title elements +\addtokomafont{titlehead}{\sffamily} +\addtokomafont{subject}{\sffamily} +\addtokomafont{title}{\sffamily} +\addtokomafont{subtitle}{\sffamily} +\addtokomafont{author}{\sffamily} +\addtokomafont{date}{\sffamily} +\addtokomafont{publishers}{\sffamily} + +% Use serif font for headings +\addtokomafont{disposition}{\rmfamily} + +% Let sections be formated as in: Aufgabe 1 -- Section title +\renewcommand*{\sectionformat}{Aufgabe \thesection\autodot\enskip--\enskip} + +% Let points of an exercise be printed as in: [__ / 2] +\newkomafont{points}{\sffamily} +\newcommand\points[2][1em]{\marginline{\framebox{{\usekomafont{points}\hspace{#1} \textbf{/} #2}}}} + +\newcounter{task} +\renewcommand{\thetask}{\@alph\c@task)} +\newenvironment{tasks} +{ + \begin{list}{\thetask} + { + \usecounter{task} + \setlength{\leftmargin}{1.6em} + } +}{% + \end{list} +} + +\DeclareNewTOC +[ + type=algorithm, + types=algorithms, + float, + floattype=4, + name=Algorithmus, + listname={Algorithmenverzeichnis} +] +{loa} diff --git a/übung_7/ngutils.sty b/übung_7/ngutils.sty new file mode 100644 index 0000000..69cc21c --- /dev/null +++ b/übung_7/ngutils.sty @@ -0,0 +1,157 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{ngutils}[2026/06/05 Never's LaTeX utils] + +\RequirePackage[ngerman]{babel} +\RequirePackage{graphicx} +\RequirePackage{tabularx} +\RequirePackage{booktabs} +\RequirePackage{listings} + +\lstnewenvironment{pseudocode}[1][] %defines the algorithm listing environment +{ + \renewcommand{\lstlistingname}{Algorithmus} + \lstset{ %this is the stype + mathescape=true, + columns=fullflexible, + basicstyle=\normalfont, + identifierstyle=\scshape, + keywordstyle=\bfseries, + keywords={, and, or, not, new, if, then, else, while, for, in, to, up, down, foreach, do, return}, + commentstyle=\itshape, + comment=[l]//, + delim=[is][\normalfont]{|}{|}, + tabsize=3, + frame=tb, + framerule=1pt, + #1 % this is to add specific settings to an usage of this environment (for instnce, the caption and referable label) + } +} +{} + +\newcommand{\seilpmi}{\Longleftarrow} + +\newcommand{\NN}{\mathbb{N}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\QQ}{\mathbb{Q}} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\PP}{\mathbb{P}} + +\newcommand{\Pe}{\mathrm{P}} +\newcommand{\NPe}{\mathrm{NP}} + +\newcommand{\Oh}{\mathcal{O}} +\newcommand{\oh}{\scriptstyle{\mathcal{O}}} + +\newcommand{\Eh}{\mathop{\mathrm{E}}} +\newcommand{\Var}{\mathop{\mathrm{Var}}} +\newcommand{\pot}{\mathop{\mathcal{P}}} +\newcommand{\argmax}{\mathop{\mathrm{arg\,max}}} +\newcommand{\argmin}{\mathop{\mathrm{arg\,min}}} + +\newcommand{\parens}[1]{\left(#1\right)} +\newcommand{\brackets}[1]{\left[#1\right]} +\newcommand{\braces}[1]{\left\{#1\right\}} +\newcommand{\angled}[1]{\left\langle#1\right\rangle} + +\newcommand{\suchthat}{\;\middle\vert\;} + +\newcommand{\tup}[1]{\parens{#1}} +\newcommand{\interval}[2][b]{ + \if\detokenize{a}\detokenize{#1}% + \left[#2\right[ + \fi + \if\detokenize{b}\detokenize{#1}% + \left[#2\right] + \fi + \if\detokenize{c}\detokenize{#1}% + \left]#2\right[ + \fi + \if\detokenize{d}\detokenize{#1}% + \left]#2\right] + \fi +} +\newcommand{\set}[2][]{ + \ifx& + \braces{#2} + \else + \braces{#1 \suchthat #2} + \fi +} +\newcommand{\arr}[1]{\angled{#1}} +\newcommand{\abs}[1]{\left\vert#1\right\vert} +\newcommand{\norm}[1]{\left\Vert#1\right\Vert} +\newcommand{\ceil}[1]{\left\lceil#1\right\rceil} +\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} + +% \newcommand{\alg}[2]{{\normalfont\scshape#1}{\normalfont(#2)}} +\newcommand{\algt}[1]{{\normalfont\scshape#1}} + +\makeatletter +\newcommand*{\alg@unstarred}[2]{{\normalfont\scshape#1}{\normalfont(#2)}} +\newcommand*{\alg@starred}[1]{{\normalfont\scshape#1}} +\newcommand*{\alg}{\@ifstar{\alg@starred}{\alg@unstarred}} +\makeatother + +\newcommand{\com}[1]{{\normalfont\itshape/\!\!/ #1}} + +\newenvironment{algorithmic}[1][1em]{ + \begin{minipage}[t]{\dimexpr\linewidth} + \bfseries + \begin{tabbing} + \hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\kill + }{% + \end{tabbing} + \end{minipage} +} + +% \newlength{\theoremskip} +% \setlength{\theoremskip}{6em} +% +% \newcommand{\theoremfont}{\normalfont\bfseries} +% \newcommand{\definitionfont}{\normalfont\bfseries} +% +% \newenvironment{theoremic}[3][]{ +% {#3 #2 #1} +% \hfill +% \begin{minipage}[t]{\the\dimexpr\linewidth-\theoremskip\relax} +% }{ +% \end{minipage} +% } +% +% \newenvironment{theorem}[1][]{ +% \begin{theoremic}[#1]{Satz}{\theoremfont} +% }{ +% \end{theoremic} +% } +% +% \newenvironment{lemma}[1][]{ +% \begin{theoremic}[#1]{Lemma}{\theoremfont} +% }{ +% \end{theoremic} +% } +% +% \newenvironment{corrolary}[1][]{ +% \begin{theoremic}[#1]{Korall}{\theoremfont} +% }{ +% \end{theoremic} +% } +% +% \newenvironment{proof}[1][]{ +% \begin{theoremic}[#1]{Beweis}{\theoremfont} +% }{ +% \end{theoremic} +% } +% +% \newenvironment{definition}[1][]{ +% \begin{theoremic}[#1]{Definition}{\theoremfont} +% }{ +% \end{theoremic} +% } +% +% \newenvironment{example}[1][]{ +% \begin{theoremic}[#1]{Beispiel}{\theoremfont} +% }{ +% \end{theoremic} +% } +%