aufgabe 2
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\section{Straßenreparatur mittels Linearer Programmierung und Flüssen}
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% vim: ft=tex
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\section{Straßenreparatur mittels Linearer Programmierung und Flüssen}
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\begin{quote}
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Gegeben sei ein ungerichteter Graph $G = \tup{V, E}$, der ein Verkehrsnetzwerk darstellt,
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wobei jeder Knoten $v \in V$ eine Stadt repräsentiert und jede Kante $e = \set{u, v}$ eine Straße
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zwischen den Städten $u$ und $v$.
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Die Straßen müssen erneuert werden. Dabei sind für jede Straße $e \in E$ Reparaturkosten
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nötig, die durch eine Funktion $r: E \to~\RR_{\geq 0}$ gegeben sind. Zusätzlich hat jede Stadt $v \in V$
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nur ein begrenztes Budget, das durch eine Funktion $B: V \to \RR_{\geq 0}$ gegeben ist.
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\end{quote}
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\begin{tasks}
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\item Seien $u_e, v_e \in \RR_{\geq 0}$ die Reparaturbeträge, die Stadt $u$ und $v$ für eine inzidente Straße
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$e$ zahlen. Um $u_e, v_e$ zu erhalten gehen wir durch die Adjazenzliste und vermerken pro Kante die inzidenten Knoten.
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Dann müssen wir folgendes lineares Programm lösen.
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Zielfunktion:
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$$
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\argmax \sum_{e \in E} u_e + v_e
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$$
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Für alle $e \in E$ gilt
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$$
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r(e) - (u_e + v_e) \leq 0
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$$
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Damit stellen wir sicher, dass die vollen Reparaturkosten bezahlt werden.
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Für alle $u \in V$ und zu $u$ inzidente Kanten $I$ gilt
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$$
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\sum_{e \in I} u_e \leq B(u)
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$$
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Damit stellen wir sicher, dass das Budget einer Stadt nicht überschritten wird.
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Wenn die Zielfunktion den Wert der Summe aller Reparaturkosten hat, haben
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wir eine Lösung gefunden.
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\points{3}
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\item Sei $x_e \in \set{0, 1}$ der Indikator, ob Stadt $x$ die Reparaturkosten der inzidenten Straße $e$ trägt. Dann müssen wir folgendes ganzahlig lineares Programm lösen.
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Zielfunktion:
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$$
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\argmax \sum_{e \in E} x_e
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$$
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Für alle $e \in E$ und deren inzidente Knoten $x, y$ gilt
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$$
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x_e + y_e \leq 1
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$$
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Damit Stellen wir sicher, dass nur eine Stadt die Reparaturkosten einer Staße trägt.
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Für alle $x \in V$ und zu $x$ inzidente Kanten $I$ gilt
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$$
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\sum_{e \in I} x_e \cdotp r(x) \leq B(x)
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$$
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Damit stellen wir sicher, dass das Budget einer Stadt nicht überschritten wird.
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Wenn die Zielfunktion den Wert der Anzahl an Knoten hat, haben wir eine
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Lösung gefunden.
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\points{2}
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\item
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Das Problem kann wie in \autoref{fig:flow} als Fluss modelliert werden.
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\begin{itemize}
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\item Dabei ist $s$ der Startknoten und $t$ der Zielknoten.
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\item Die Kantenkapazitäten von $s$ zu den Städten $v_i \in V$ ist das Budget $B(v_i)$.
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\item Die Kantenkapazitäten von $v_i$ zu den Straßen $e_j \in E$ müssen jeweils
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größer als $r(e_j)$ sein. Jede Straße $e_j$ hat genau zwei eingehende Kanten von
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den beiden inzidenten Städten.
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\item Die Kantenkapazitäten von $e_j$ zu $t$ sind die Reparaturkosten $r(e_j)$
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\end{itemize}
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Wenn ein zulässiger Fluss alle in $t$ eingehenden Kanten füllt, dann gibt es
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eine Lösung für das Problem.
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Das Modell ist korrekt, da
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\begin{itemize}
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\item Die Beiträge, die eine Stadt $v_i$ leisten kann von den
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eingehenden Kantenkapazitäten limitiert werden.
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\item Die Straßen $e_j$ genau zwei eingehende Kanten haben.
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\item Die Reparaturkosten einer Straße $e_j$ von den ausgehenden Kantenkapazitäten
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limitiert werden.
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\end{itemize}
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\points{4}
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\begin{figure}[h]
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\centering
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\includegraphics[page=2, width=0.8\textwidth]{figures.pdf}
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\caption{Das Problem als Flussproblem. Die blauen Knoten sind Städte und die orangenen Straßen.}
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\label{fig:flow}
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\end{figure}
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\end{tasks}
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110 96 s
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100 204 l
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480 160
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<text matrix="1 0 0 1 0 8" pos="448 128" stroke="black" type="label" width="19.416" height="9.0144" depth="3.024" halign="center" valign="bottom" size="tiny" style="math">3/3</text>
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<text matrix="1 0 0 1 0 -16" pos="384 192" stroke="black" type="label" width="19.416" height="9.0144" depth="3.024" halign="center" valign="bottom" size="tiny" style="math">2/2</text>
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<text matrix="0 -1 1 0 68 576" pos="320 256" stroke="black" type="label" width="19.416" height="9.0144" depth="3.024" halign="center" valign="bottom" size="tiny" style="math">0/3</text>
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<text matrix="1 0 0 1 0 4" pos="448 384" stroke="black" type="label" width="19.416" height="9.0144" depth="3.024" halign="center" valign="bottom" size="tiny" style="math">6/6</text>
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448 320 m
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576 256 l
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</path>
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448 256 m
|
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576 256 l
|
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</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
448 192 m
|
||||
576 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
192 256 m
|
||||
320 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
192 256 m
|
||||
320 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
192 256 m
|
||||
320 192 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 256 m
|
||||
448 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 256 m
|
||||
448 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 192 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 192 m
|
||||
448 192 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 256 l
|
||||
</path>
|
||||
<use name="mark/disk(sx)" pos="192 256" size="normal" stroke="black"/>
|
||||
<use name="mark/disk(sx)" pos="320 320" size="normal" stroke="dark blue"/>
|
||||
<use name="mark/disk(sx)" pos="320 256" size="normal" stroke="dark blue"/>
|
||||
<use name="mark/disk(sx)" pos="320 192" size="normal" stroke="dark blue"/>
|
||||
<use name="mark/disk(sx)" pos="448 320" size="normal" stroke="dark orange"/>
|
||||
<use name="mark/disk(sx)" pos="448 256" size="normal" stroke="dark orange"/>
|
||||
<use name="mark/disk(sx)" pos="448 192" size="normal" stroke="dark orange"/>
|
||||
<use name="mark/disk(sx)" pos="576 256" size="normal" stroke="black"/>
|
||||
<text matrix="1 0 0 1 0 8" transformations="translations" pos="320 320" stroke="black" type="label" width="18.7176" height="8.5848" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">v_1</text>
|
||||
<text matrix="1 0 0 1 0 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="18.7176" height="8.5848" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">v_2</text>
|
||||
<text matrix="1 0 0 1 4 -136" transformations="translations" pos="320 320" stroke="black" type="label" width="19.764" height="8.604" depth="4.824" halign="center" valign="top" size="footnote" style="math">v_n</text>
|
||||
<text matrix="1 0 0 1 128 8" transformations="translations" pos="320 320" stroke="black" type="label" width="17.9904" height="8.8152" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">e_1</text>
|
||||
<text matrix="1 0 0 1 -128 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="7.7664" height="8.8368" depth="0.192" halign="center" valign="bottom" size="footnote" style="math">s</text>
|
||||
<text matrix="1 0 0 1 128 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="17.9904" height="8.8152" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">e_2</text>
|
||||
<text matrix="1 0 0 1 128 -136" transformations="translations" pos="320 320" stroke="black" type="label" width="21.6624" height="8.8344" depth="4.824" halign="center" valign="top" size="footnote" style="math">e_m</text>
|
||||
<text matrix="1 0 0 1 256 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="8.34" height="10.9416" depth="0.192" halign="center" valign="bottom" size="footnote" style="math">t</text>
|
||||
<text matrix="0.894704 0.446659 -0.446659 0.894704 112.622 -137.236" pos="320 320" stroke="black" type="label" width="33.0312" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">B(v_1)</text>
|
||||
<text matrix="0.894427 -0.447214 0.447214 0.894427 -73.1492 134.135" pos="256 224" stroke="black" type="label" width="33.9288" height="8.9664" depth="3.048" halign="center" valign="top" size="tiny" style="math">B(v_n)</text>
|
||||
<text transformations="translations" pos="256 260" stroke="black" type="label" width="33.0312" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">B(v_2)</text>
|
||||
<text matrix="0.894427 -0.447214 0.447214 0.894427 186.851 218.135" pos="256 224" stroke="black" type="label" width="28.2504" height="8.976" depth="3.984" halign="center" valign="top" size="tiny" style="math">r(e_1)</text>
|
||||
<text matrix="0.894704 0.446659 -0.446659 0.894704 368.622 -209.236" pos="320 320" stroke="black" type="label" width="31.6704" height="8.9664" depth="3.048" halign="center" valign="top" size="tiny" style="math">r(e_m)</text>
|
||||
<text matrix="1 0 0 1 256 0" transformations="translations" pos="256 260" stroke="black" type="label" width="28.2504" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">r(e_2)</text>
|
||||
<text transformations="translations" pos="320 224" stroke="black" type="label" width="5.3184" height="10.98" depth="1.32" halign="center" valign="center" size="footnote" style="math">\vdots</text>
|
||||
<text matrix="1 0 0 1 128 0" transformations="translations" pos="320 224" stroke="black" type="label" width="5.3184" height="10.98" depth="1.32" halign="center" valign="center" size="footnote" style="math">\vdots</text>
|
||||
</page>
|
||||
</ipe>
|
||||
|
|
@ -42,6 +42,8 @@
|
|||
\newcommand\oh{\ensuremath{\scriptstyle{\mathcal{O}}}}
|
||||
\newcommand\Eh{\ensuremath{\mathrm{E}}}
|
||||
\newcommand\Var{\ensuremath{\mathrm{Var}}}
|
||||
\newcommand\argmax{\ensuremath{\mathrm{arg\,max}}}
|
||||
\newcommand\argmin{\ensuremath{\mathrm{arg\,min}}}
|
||||
|
||||
\newcommand\parens[1]{\ensuremath{\left(#1\right)}}
|
||||
\newcommand\brackets[1]{\ensuremath{\left[#1\right]}}
|
||||
|
|
@ -61,6 +63,7 @@
|
|||
\newcommand\algt[1]{{\normalfont\scshape#1}}
|
||||
\newcommand\com[1]{{\normalfont\itshape/\!\!/ #1}}
|
||||
|
||||
|
||||
\newenvironment{algorithmic}[1][1em]{
|
||||
\begin{minipage}[t]{\dimexpr\linewidth}
|
||||
\bfseries
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue