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Übungsblatt} +\subject{Algorithmische Graphentheorie} +\author{Jasper Gude \and Pia Röttgers} + +\begin{document} +\maketitle +\points[\qquad]{20} + +\input{aufgabe_1.tex} +\input{aufgabe_2.tex} +\input{aufgabe_3.tex} + +\end{document} diff --git a/übung_4/aufgabe_1.tex b/übung_4/aufgabe_1.tex new file mode 100644 index 0000000..62f2623 --- /dev/null +++ b/übung_4/aufgabe_1.tex @@ -0,0 +1,33 @@ +\section{Knotengrade} +\begin{tasks} +\item + Einen solchen Graph gibt es. Siehe \autoref{fig:task1a}. + \points{3} + \begin{figure} + \centering + \includegraphics[page=1, width=0.8\textwidth]{figures.pdf} + \caption{Ein Graph mit 11 Knoten und Knotengraden 1, 1, 2, 3, 4, 5, 5, 6, 7, 8, 10.} + \label{fig:task1a} + \end{figure} +\item + Das Problem kann als Maximalflussproblem modelliert werden. Dafür haben wir + eine Quelle $s$, eine Senke $t$ und Knoten $v_i$ mit $i \in \NN_{\leq n}$. + Die Eingangskapazität eines Knoten $v_i$ entspricht dem Eingangsgrad $e_i$. + Die Ausgangskapazität entspricht dem Ausgangsgrad $a_i$. Siehe \autoref{fig:task1b}. + + Wenn es einen vollständigen Fluss gibt, gibt es eine Lösung für das Problem, + sonst nicht. + \points{4} + \begin{figure} + \centering + \includegraphics[page=2, width=0.5\textwidth]{figures.pdf} + \caption{Das Problem als Maximalflussproblem.} + \label{fig:task1b} + \end{figure} +\item + Die Modellierung ist korrekt, da der Eingangs- und Ausgangsgrad jedes Knotens $v_i$ + von oben beschränkt wird von der Eingangskapazität $e_i$ bzw. Ausgangskapazität $a_i$. + + Zudem erfüllt die Flusserhaltung den Zweck, dass die Summe der Eingangsgrade gleich der Summe der Ausgangsgrade sind, was in einem Graphen erfüllt sein muss. + \points{2} +\end{tasks} diff --git a/übung_4/aufgabe_2.tex b/übung_4/aufgabe_2.tex new file mode 100644 index 0000000..f821048 --- /dev/null +++ b/übung_4/aufgabe_2.tex @@ -0,0 +1,5 @@ +\section{b-Flüsse} +\begin{tasks} +\item +\item +\end{tasks} diff --git a/übung_4/aufgabe_3.tex b/übung_4/aufgabe_3.tex new file mode 100644 index 0000000..78a1258 --- /dev/null +++ b/übung_4/aufgabe_3.tex @@ -0,0 +1 @@ +\section{Minimale Schnitte} diff --git a/übung_4/aufgabe_4.tex b/übung_4/aufgabe_4.tex new file mode 100644 index 0000000..4535a11 --- /dev/null +++ b/übung_4/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. Da wir nicht von $\Pe = \NPe$ ausgehen, + ist das nicht möglich. + \points{2} + +\end{tasks} diff --git a/übung_4/figures.pdf b/übung_4/figures.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9236211972a91a785da05261a11e0cccf69e69f1 GIT binary patch literal 22397 zcmeHv2V7Li(l3%3q69$+4oZ%bXAnd(k~4w?NkfpFvw76Wcc1QmS65Y6)oE@`IR#;;2oi(_L%_}! zr@;~uqB@=y9=>NC>Zopp$a}TT>*3#JusH2K?vhlD5 zLoqNhNlB2qha1-15#%!vsOPV#bK*cdcln#+hiye%=dLMQm8lz%fE>%R>zHpIIwYXW zc+xk4pe6xKu|#!ClOmuYLx8vmE)Uvw8B`+CM2KZ0Bd&V@I%%3)a51v?|PVg+6jy{Q)LLI#SG71~loa_joBV`-jV$M%XR*?6g>b*)Hnkf0{K zo882X3L7H6FlI$=m_>JAG@Yan_F-&Kovi!9;2c)JbS8aj^Td}xW;Amdx^m>&qbGBv zZT(*zz0vRXJy6fkKb+1%G`<9##cqKFc2RA-&aTaZbJcBUtE{-irXMEXwnYM)C__hw+wCdJD` zeR2W;Z$S^#@2X5J`UFv1Xt(fnN?TcLjgNBvCYVoT6KIC_l zUZ1_!o5Y~t@|0Zt0DYNo1J7c=Qa_i+z&^67!`G`b`~C zqT5^U{}5BXVf~~$Biy$X+17V(Bb@wW-F{K`Prl{n43noN#^xCUY(Ip(c;6A5nn$77 z$!(Tu^0nCFLGh`Cd~d0~Q)erCU}K-`KPDZ18K^38^uc>nw11OCJ*oTLlg)F_e9C8N 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+\setlength{\fboxsep}{1pt} + +\newcommand{\bk}{\color{black}} +\newcommand{\wt}{\color{white}} +\newcommand{\bu}{\color{dark blue}} +\newcommand{\bn}{\color{dark brown}} +\newcommand{\cn}{\color{dark cyan}} +\newcommand{\gy}{\color{dark gray}} +\newcommand{\gn}{\color{dark green}} +\newcommand{\og}{\color{dark orange}} +\newcommand{\pk}{\color{dark pink}} +\newcommand{\pl}{\color{dark purple}} +\newcommand{\rd}{\color{dark red}} +\newcommand{\yo}{\color{dark yellow}} +\newcommand{\tl}{\color{title}} + + + + + + + + + + + + + + + + + + + + +384 384 m +480 416 l + + +384 384 m +288 416 l + + +384 384 m +480 160 l + + +384 384 m +192 256 l + + +384 384 m +384 128 l + + +384 384 m +192 128 l + + +384 384 m +576 128 l + + +384 384 m +192 384 l + + +384 384 m +480 352 l + + +384 384 m +576 384 l + + +480 160 m +192 256 l + + +480 160 m +384 128 l + + +480 160 m +192 128 l + + +480 160 m +576 128 l + + +480 160 m +192 384 l + + +480 160 m +480 352 l + + +480 160 m +576 384 l + + +384 128 m 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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}{\normalfont \sffamily \itshape} +\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][\quad]{\marginline{\framebox{{\usekomafont{points}#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_4/ngutils.sty b/übung_4/ngutils.sty new file mode 100644 index 0000000..7fb21db --- /dev/null +++ b/übung_4/ngutils.sty @@ -0,0 +1,124 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{ngutils}[2025/12/24 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={, 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{\tup}[1]{\parens{#1}} +\newcommand{\ivl}[1]{\brackets{#1}} +\newcommand{\set}[1]{\braces{#1}} +\newcommand{\arr}[1]{\angled{#1}} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\ceil}[1]{\lceil#1\rceil} +\newcommand{\floor}[1]{\lfloor#1\rfloor} + +\newcommand{\alg}[2]{{\normalfont\scshape#1}{\normalfont(#2)}} +\newcommand{\algt}[1]{{\normalfont\scshape#1}} +\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}[2][\theoremfont]{ + {#1 #2} + \hfill + \begin{minipage}[t]{\the\dimexpr\linewidth-\theoremskip\relax} + }{ + \end{minipage} +} + +\newenvironment{theorem}{ + \begin{theoremic}{Satz.} + }{ + \end{theoremic} +} + +\newenvironment{lemma}{ + \begin{theoremic}{Lemma.} + }{ + \end{theoremic} +} + +\newenvironment{proof}{ + \begin{theoremic}{Beweis.} + }{ + \end{theoremic} +} + +\newenvironment{definition}{ + \begin{theoremic}[\definitionfont]{Definition.} + }{ + \end{theoremic} +} + +\newenvironment{example}{ + \begin{theoremic}[\definitionfont]{Beispiel.} + }{ + \end{theoremic} +} + +