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Folglich kann eine konstante Gütegarantie nur für vergleichsweise wenig Probleme erreicht werden. +\section{Relative Gütegarantie} +\subsection{Definition} +\begin{enumerate} + \item $A$ hat bei Eingabe $I$ eine relative Güte von + \begin{equation*} + \rho_A(I) = \max\left\{\frac{A(I)}{\opt(I)}, \frac{\opt(I)}{A(i)}\right\} \ge 1 + \end{equation*} + \item Die relative worst-case-Güte von $A$ ist die Funktion + \begin{equation*} + \rho_A^\mathrm{wc}(n) = \max\left\{\rho_A(I)\mid I \in \mathcal{D}, \abs{i}\le n\right\} + \end{equation*} + \item $A$ garantiert eine relative Güte von $\rho_A : \mathbb{N} \mapsto \mathbb{N}$, falls für alle $n \in \mathbb{N}$ gilt + \begin{equation*} + \rho_A^{\mathrm{wc}}(n) \le \rho_A(n) + \end{equation*} + \item $A$ macht für die Eingabe $I \in \mathcal{D}$ einen relativen Fehler von + \begin{equation*} + \varepsilon_A(I) = \frac{\abs{A(I) - \opt(I)}}{\opt(I)} = \abs{\frac{A(I)}{\opt(I)} - 1} + \end{equation*} + \item $A$ garantiert einen relativen Fehler von $\varepsilon_A(n)$, falls für alle $\{I\mid I \in \mathcal{D}, \abs{I}\le n\}$ gilt + \begin{equation*} + \varepsilon_A(I) \le \varepsilon_A(n) + \end{equation*} + \item $A$ hat eine relative Abweichung von $\rho'_A : \mathbb{N} \mapsto \mathbb{N}$, falls für unendlich viele $n$ gilt + \begin{equation*} + \rho_A^\mathrm{wc}(n) \ge \rho'_A(n) + \end{equation*} + + Eine unendlich große Menge $\mathcal{D}' \subseteq \mathcal{D}$ heißt $\rho'_A(n)$-Zeugenmenge gegen $A$, wenn für alle $I \in \mathcal{D}'$ gilt + \begin{equation*} + \rho_A(I) \ge \rho'_A(\abs{I}) + \end{equation*} +\end{enumerate} -\section{Graphfärbbarkeit} +Es folgen daraus direkt, dass +\begin{enumerate} + \item bei einem Minimierungsproblem $1 + \varepsilon_A(n) = \rho_A(n)$ ist. + \item bei einem Maximierungsproblem $1 - \varepsilon_A(n) = \frac{1}{\rho_A(n)}$ ist. + \item für alle Probleme $\varepsilon_A(n) \le \rho_A(n) - 1$ ist. +\end{enumerate} + +Weiter lassen sich damit obere bzw. untere Schranken der Optimallösung aus einer approximierten Lösung angeben. Es folgt, dass +\begin{enumerate} + \item bei einem Minimierungsproblem gilt + \begin{equation*} + \frac{1}{\rho_A(\abs{I})} \cdot A(I) \le \opt(I) \le A(I) \le \rho_A(\abs{I})\cdot \opt(I) + \end{equation*} + \item bei einem Maximierungsproblem gilt + \begin{equation*} + \frac{1}{\rho_A(\abs{I})} \cdot \opt(I) \le A(I) \le \opt(I) \le \rho_A(\abs{I}) \cdot A(I) + \end{equation*} + \item bei beiden Problemtypen mit der Beziehung + \begin{equation*} + \abs{A(I) - \opt(I)} \le \varepsilon_A(\abs{I}) \cdot \opt(I) + \end{equation*} + gilt + \begin{equation*} + (1 - \varepsilon_A(\abs{I})) \cdot \opt(I) \le A(I) \le (1 + \varepsilon_A(\abs{I})) \cdot \opt(I) + \end{equation*} +\end{enumerate} + +\section{Das Problem der Unabhängigen Knotenmengen \problem{IS}} + +\section{Graphfärbbarkeitsprobleme} \subsection{Knotenfärbungsproblem} \subsubsection{Definition} \begin{align*} @@ -176,67 +238,6 @@ Die Größe der kleinsten möglichen Kantenfärbung ist der chromatische Index $ \subsubsection{Algorithmen} TODO: Übung -\section{Relative Gütegarantie} -\subsection{Definition} -\begin{enumerate} - \item $A$ hat bei Eingabe $I$ eine relative Güte von - \begin{equation*} - \rho_A(I) = \max\left\{\frac{A(I)}{\opt(I)}, \frac{\opt(I)}{A(i)}\right\} \ge 1 - \end{equation*} - \item Die relative worst-case-Güte von $A$ ist die Funktion - \begin{equation*} - \rho_A^\mathrm{wc}(n) = \max\left\{\rho_A(I)\mid I \in \mathcal{D}, \abs{i}\le n\right\} - \end{equation*} - \item $A$ garantiert eine relative Güte von $\rho_A : \mathbb{N} \mapsto \mathbb{N}$, falls für alle $n \in \mathbb{N}$ gilt - \begin{equation*} - \rho_A^{\mathrm{wc}}(n) \le \rho_A(n) - \end{equation*} - \item $A$ macht für die Eingabe $I \in \mathcal{D}$ einen relativen Fehler von - \begin{equation*} - \varepsilon_A(I) = \frac{\abs{A(I) - \opt(I)}}{\opt(I)} = \abs{\frac{A(I)}{\opt(I)} - 1} - \end{equation*} - \item $A$ garantiert einen relativen Fehler von $\varepsilon_A(n)$, falls für alle $\{I\mid I \in \mathcal{D}, \abs{I}\le n\}$ gilt - \begin{equation*} - \varepsilon_A(I) \le \varepsilon_A(n) - \end{equation*} - \item $A$ hat eine relative Abweichung von $\rho'_A : \mathbb{N} \mapsto \mathbb{N}$, falls für unendlich viele $n$ gilt - \begin{equation*} - \rho_A^\mathrm{wc}(n) \ge \rho'_A(n) - \end{equation*} - - Eine unendlich große Menge $\mathcal{D}' \subseteq \mathcal{D}$ heißt $\rho'_A(n)$-Zeugenmenge gegen $A$, wenn für alle $I \in \mathcal{D}'$ gilt - \begin{equation*} - \rho_A(I) \ge \rho'_A(\abs{I}) - \end{equation*} -\end{enumerate} - -Es folgen daraus direkt, dass -\begin{enumerate} - \item bei einem Minimierungsproblem $1 + \varepsilon_A(n) = \rho_A(n)$ ist. - \item bei einem Maximierungsproblem $1 - \varepsilon_A(n) = \frac{1}{\rho_A(n)}$ ist. - \item für alle Probleme $\varepsilon_A(n) \le \rho_A(n) - 1$ ist. -\end{enumerate} - -Weiter lassen sich damit obere bzw. untere Schranken der Optimallösung aus einer approximierten Lösung angeben. Es folgt, dass -\begin{enumerate} - \item bei einem Minimierungsproblem gilt - \begin{equation*} - \frac{1}{\rho_A(\abs{I})} \cdot A(I) \le \opt(I) \le A(I) \le \rho_A(\abs{I})\cdot \opt(I) - \end{equation*} - \item bei einem Maximierungsproblem gilt - \begin{equation*} - \frac{1}{\rho_A(\abs{I})} \cdot \opt(I) \le A(I) \le \opt(I) \le \rho_A(\abs{I}) \cdot A(I) - \end{equation*} - \item bei beiden Problemtypen mit der Beziehung - \begin{equation*} - \abs{A(I) - \opt(I)} \le \varepsilon_A(\abs{I}) \cdot \opt(I) - \end{equation*} - gilt - \begin{equation*} - (1 - \varepsilon_A(\abs{I})) \cdot \opt(I) \le A(I) \le (1 + \varepsilon_A(\abs{I})) \cdot \opt(I) - \end{equation*} -\end{enumerate} - \section{Das metrische Traveling Salesperson Problem $\Delta\problem{TSP}$} \subsection{Definition} \begin{align*}