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za5>4Q#xsMw@M^yn$!7ScD=J$O*IVz;LkTH02P2jvQHnW`W9GGxCg-F}NTV~kD?To$ zDk>Q{J#JAbr6uY{zC(X7E1UgafO?xRR*uLXo|ZsQU!biWGOqw1KcCPCQbSH%9{Ilj Dsp@_# diff --git a/zusammenfassung.tex b/zusammenfassung.tex index 2c9f6dc..4bf30cf 100644 --- a/zusammenfassung.tex +++ b/zusammenfassung.tex @@ -59,6 +59,7 @@ \newcommand{\greedycol}{\algo{GreedyCol}} \newcommand{\greedycoltwo}{\algo{GreedyCol2}} \newcommand{\greedycoledge}{\algo{GreedyColEdge}} +\newcommand{\greedyplanarcol}{\algo{GreedyPlanarCol}} \newcommand{\vertices}{\mathrm{V}} \newcommand{\edges}{\mathrm{E}} \newcommand{\degree}{\mathrm{deg}} @@ -242,6 +243,39 @@ Eine äquivalente Charakterisierung ist: Das NP-vollständige Entscheidungsprobl \item \rucksack{} ist schwach NP-vollständig, weil es einen pseudo-polynomiellen Algorithmus für die Optimierungsvariante gibt, der auch für das Entscheidungsproblem verwendet werden kann. \end{itemize} +\subsection{Planarer Graph} +Ein planarer Graph kann so auf einer Kugel dargestellt werden, dass sich keine seiner Kanten kreuzen. +\begin{satz}[Eulerscher Polyedersatz] + Für einen beliebigen zusammenhängenden planaren Graph mit $n$ Knoten, $m$ Kanten und $f$ Facetten gilt: + \begin{equation*} + n - m + f = 2 + \end{equation*} +\end{satz} +\begin{proof}[Beweis durch Induktion] + \begin{itemize} + \item Induktionsanfang: Für einen Knoten gilt $1 - 0 + 1 =2$. + \item Induktionsschritt: Fallunterscheidung:\begin{itemize} + \item Hinzufügen einer Ecke: Die Ecke wird mit einer bestehenden Kante verbunden (zusammenhängender Graph), also gilt $n + 1 - (m + 1) + f = n - m +f = 2$. + \item Hinzufügen einer Kante: Eine bestehende Fläche wird in zwei geteilt, also gilt $n - (m + 1) + f + 1 = n - m + f = 2$. + \end{itemize} + \end{itemize} +\end{proof} + +\begin{satz} + Für einen planaren Graph gilt + \begin{equation*} + m \le 3\cdot n - 6 + \end{equation*} +\end{satz} + +\begin{satz} + Ein planarer Graph enthält mindestens einen Knoten mit Grad 5 oder weniger. +\end{satz} +\begin{proof}[Beweis durch Widerspruch] + Nähme man an, dass einen planaren Graphen mit lediglich Knoten vom Grad 6 oder höher gibt, muss dieser Graph $m = 3\cdot n$ Knoten haben. + Es folgt dann der Widerspruch $n = 3\cdot m \not\le 3\cdot m - 6$. +\end{proof} + \section{Absolute Gütegarantie} \subsection{Definition} \begin{enumerate} @@ -502,6 +536,29 @@ Sei $\Pi$ ein Optimierungsproblem und $A$ ein Approximationsalgorithmus für $\P $\Delta(G) - 1$-Zeuge gegen \greedycol: TODO (Abbildung 2.1) \end{zeuge} +\subsection{\greedyplanarcol} +\begin{algorithmic}[1] + \If{$G$ ist knoten-2-färbbar} + \State \Return berechne Knoten-2-Färbung + \EndIf + \State wähle Knoten $u$ mit höchstem Grad $\Gamma(u)$ \Comment{$\Gamma(u) \le 5$} + \State entferne $u$ und seine $\Gamma(u) \le 5$ Kanten aus $G$ + \ForAll{übrige Teilgraphen $G_i$} \Comment{$1 \le i \le \Gamma(u)$} + \State $c_\vertices = c_\vertices \cup \greedyplanarcol(G_i)$ + \EndFor + \State $c_\vertices(u) =$ kleinste der freien Farbe aus $\{1, \dots, 6\}$ + \State \Return $c_\vertices$ +\end{algorithmic} +\begin{satz} + \greedyplanarcol{} hat eine maximale Gütegarantie von $\kappa_\greedyplanarcol(n) = 3$. +\end{satz} +\begin{proof} + Ist $G$ knoten-2-färbbar, so liefert \greedyplanarcol{} auch eine Knoten-2-Färbung zurück. Es gilt also im Folgenden $\opt(G) \ge 3$ und damit + \begin{equation*} + \kappa_\greedyplanarcol(n) = \abs{\greedyplanarcol(G) - \opt(G)} \le \abs{6 - 3} = 3 + \end{equation*} +\end{proof} + \subsection{\greedycoltwo} \begin{algorithmic}[1] \State $t = 1, V^{(1)} = V$