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zm9eDOr4*;xL7YEqSsCv-%>VBG1~qLGoSqJI9XEET_j>v#+=x-N)jVeP;)tM|w#^ge?t4L-w;u lc~1xwLcpm1vst*hSwP&pV2&p$Hy1yL04hDbjG8R!e*ubQXr%xE literal 0 HcmV?d00001 diff --git a/übung_9/agt_übung_9.tex b/übung_9/agt_übung_9.tex index 3a2edd9..88c2f9c 100644 --- a/übung_9/agt_übung_9.tex +++ b/übung_9/agt_übung_9.tex @@ -4,12 +4,12 @@ \setkeys{Gin}{pagebox=artbox, width=0.8\textwidth} \subject{Algorithmische Graphentheorie} -\title{8. Übungsblatt} +\title{9. Übungsblatt} \author{Jasper Gude \and Pia Röttgers} \begin{document} \maketitle -\points[2em]{20} +\points[2em]{34} \input{aufgabe_1.tex} \input{aufgabe_2.tex} diff --git a/übung_9/aufgabe_1.tex b/übung_9/aufgabe_1.tex index 0bfc858..47c9b82 100644 --- a/übung_9/aufgabe_1.tex +++ b/übung_9/aufgabe_1.tex @@ -1,23 +1,50 @@ -\section{Kleinste Schnitte} +\section{Dreifärbarkeit} \begin{tasks} \item Siehe \autoref{fig:1a}. - \points{2} + \points{1} \begin{figure} \centering \includegraphics[page=1, width=0.5\textwidth]{figures.pdf} - \caption{Gegenbeispiel; Minimaler Schnitt $\tup{S, T}$ und nicht minimaler Schnitt $\tup{\set{v}, V \setminus \set{v}}$ in rot.} + \caption{Dreifärbung des Sterngraphs.} \label{fig:1a} - \end{figure} \item - Die Wahrscheinlichkeit, dass \alg*{Contract} in keiner Iteration - eine Kante aus $C = \set{uv \in E \mid u \in S, v \in T}$ mit minimalem Schnitt $\tup{S, T}$ kontrahiert, also immer die falschen Knoten auswählt, - ist laut Vorlesung $\frac{2}{n(n-1)}$. Das heißt die Zahl der - richtigen Knoten wächst quadratisch, da die Wahrscheinlichkeit quadratisch - abnimmt. - \points{3} + Da die Knoten, die zwei Türme miteinander verbinden, die selbe Farbe haben, + kann man zwischen zwei Türme einen weiteren Turm einfügen, der die + Färbbarkeitsregeln nicht verletzt. + + Die Knoten mit Grad 2 im Sterngraphen sind die Spitzen der Türme. + Da die Turmverbindungsknoten alle die selbe Farbe haben müssen und die + zu ihnen ajazenten Knoten jeweils unterschiedlich gefärbt sein müssen, + da sie selbst adjazent zueinander sind, müssen die Spitzen die letzte + freie Farbe bekommen. + \points{2} + \item + Um das Problem 3COL auf das Problem 3COL4 zu reduzieren, müssen wir + dafür sorgen, dass Knoten mit Grad größer $4$ so aufgelöst werden, dass + sie maximal Grad $4$ haben. + + Dazu machen wir uns zu nutze, dass der Sterngraph beliebig erweiterbar + ist und somit beliebig viele Knoten mit Grad $2$ hat, die alle die + selbe Farbe haben müssen. + + Einen Knoten $v$ mit Grad $m > 4$ ersetzen wir durch einen Stern mit $m$ + Zacken. Die zu $v$ adjazenten Kanten verbinden wir mit den Spitzen des + Sterns. + + Jetzt haben wir einen Graphen mit Maximalgrad $4$ auf dem wir \alg*{Test3COL4} + anwenden können. + + Beim Rückübersetzen können wir die Sterne wieder durch einen einzigen Knoten + ersetzen, der die Farbe der Spitzen hat. Da diese die selbe Farbe haben, + verletzt das nicht die Färbbarkeitsregel. + + Da 3COL NP-vollständig ist, muss also 3COL4 und insbesondere auch \alg*{Test3COL4} + auch NP-vollständig sein. + \points{4} + \end{tasks} diff --git a/übung_9/aufgabe_2.tex b/übung_9/aufgabe_2.tex index 84af509..660a148 100644 --- a/übung_9/aufgabe_2.tex +++ b/übung_9/aufgabe_2.tex @@ -1,25 +1,29 @@ -\section{Implementierung von \textsc{Contract}} -\begin{enumerate} - \item $G$ nach $H$ kopieren. \hfill $\Oh(1)$ - \item Wenn $\abs{V_H} \leq 2$, dann ist die Zerlegung $\tup{S, T}$ von $G$, - die den beiden letzen Knoten in $H$ entspricht, das Ergebnis. - \label{step2} - \item Wähle eine Zufallszahl $z$ im Intervall $\interval{1; \abs{V_H}}$. \hfill $\Oh(1)$ - \item Nimm den Knoten $a = V_H[z]$ und wähle eine Zufallszahl $z'$ im Intervall $\interval{1, \abs{Adj[a]}}$. +\section{Chromatische Zahl} +\begin{tasks} + \item + Angenommen der Graph $G$ ist vollständig. So hat jeder Knoten den maximalen + Knotengrad $\Delta(G)$. Für alle Knoten $v$ gilt, dass jeder adjazente + Knoten und $v$ eine eigene Farbe haben muss. Das sind also $\Delta(G) + 1$ + viele. Jeder Graph (mit weniger Kanten) hat also eine chromatische Zahl + $\chi(G) \leq \Delta(G) + 1$. + \points{2} + + \item + Für alle Anzahlen an Knoten $n \in \NN$ und Maximalgrad $\Delta < n$ existiert ein + Graph $G_{n,\Delta}$ mit $\chi(G_{n,\Delta}) = \Delta + 1$. Dieser Graph + hat eine Clique der Größe $\Delta$. + \points{2} + + \item + \begin{align*} + & \text{Variablen:} & x_1, \dots, x_{n} \in \set{1, \dots, \Delta + 1} \\ + &&\text{Jeder Knoten hat eine Farbe} \\ + & \text{Zielfunktion:} & \argmin \sum_{i=1}^{n} x_i \\ + &&\text{Es werden so wenig Farben wie möglich benutzt} \\ + & \text{Nebenbedingung:} & \forall ab \colon x_a \neq x_b \\ + &&\text{Die Knoten einer Kante haben nicht die selbe Farbe} + \end{align*} + \points{2} - Nimm den Knoten $b = Adj_{z'}[a]$. \hfill $\Oh(1)$ - \item Bestimme für jeden zu $a$ oder $b$ adjazenten Knoten $c_i$ die Anzahl der Kanten - zwischen $c_i$ und $a$ oder $b$. \hfill $\Oh(V_H) = \Oh(n)$. - \item Kontrahiere die Kante $ab$. Lösche dazu die Knoten $a, b$ sowie alle zu $a$ oder $b$ inzidenten Kanten. Da Mehrfachkanten als Zahl implementiert sind, sind - nur maximal 2 Einträge pro $c_i$ zu löschen. \hfill $\Oh(n)$ - - Füge einen neuen Knoten $d$ ein. Füge für jeden Knoten $c_i$ die vorher bestimmte Anzahl - an Kanten zwischen $c_i$ und $a$ oder $b$ als Kanten zwischen $c_i$ und $d$ ein. - \hfill $\Oh(n)$ - \item Gehe zurück zu \autoref{step2} -\end{enumerate} - -Da der Knoten $a$ durch eine gleichverteilte Zufallszahl $z$ ausgewählt wird und der -Knoten $b$ ebenfalls gleichverteilt ausgewählt wird, ist die Kante $ab$ ebenfalls -gleichverteilt ausgewählt. -\points{6} + +\end{tasks} diff --git a/übung_9/figures.pdf b/übung_9/figures.pdf index cebf2b1ef2356a8589f1ba577b11beff756967b6..55e927484112afc033d40a7fc70dfa8697a6bbfb 100644 GIT binary patch delta 6176 zcmZWpWl$6hx29_skPzuo5Rlkq>2B$c1*E&XktJ8US)@S`1O%4uP6_D-SwK=kN*eC_ zes|{n`0n}f%;}gp=X7bq&5$01n7kmtG%&?0n2~tmp#boBm%K`c@-{O#%3C@SDTRTx z#O&btkW8)=Lmf1~Z|d9R@1}N3cBA!)-H5*Fq5bA4X~-;R{rvAlpIevYCQ0M$^4~|r zPG{lzqluyFBr~AGF&e9|Oo90|k>xV{DIZ+J(t2KHssW0a0GJ)gibf>)KaVe5{h2rp zJNk)z`?UC|aH7uDr}sDL!Xd75dOiKL_31rkn=4PHw~Gs=iKwX5q#kqnC!IY36wk8x 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