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b/zusammenfassung.tex @@ -86,7 +86,9 @@ \newcommand{\setcover}{\problem{SetCover}} \newcommand{\cov}{\mathrm{cov}} \newcommand{\detroundsc}{\algo{DetRoundSC}} -\newcommand{\randroundingscr}{\algo{RandRoundingSC}[r]} +\newcommand{\randroundscr}{\algo{RandRoundSC}[r]} +\newcommand{\lasvegassc}{\algo{LasVegasSC}} +\newcommand{\lasvegasscr}{\lasvegassc[r]} % Beweisumgebungen \newtheorem*{satz}{Satz} @@ -940,7 +942,7 @@ Damit lautet das ILP $X$ für \setcover: \end{equation*} \end{proof} -\subsection{Unzuverlässiges Randomized Rounding mit $\randroundingscr$} +\subsection{Unzuverlässiges Randomized Rounding mit $\randroundscr$} \begin{algorithmic}[1] \State $(x_1, \dots, x_m) =$ löse $X_\rel$ \State $\chi = \emptyset$ @@ -949,10 +951,10 @@ Damit lautet das ILP $X$ für \setcover: \EndFor \State \Return $\chi$ \end{algorithmic} -\randroundingscr{} ist ein sogenannter Monte-Carlo-Algorithmus, weil er auch nicht zulässige Lösungen zurückliefern kann. +\randroundscr{} ist ein sogenannter Monte-Carlo-Algorithmus, weil er auch nicht zulässige Lösungen zurückliefern kann. \begin{satz} - Für eine Eingabe $S$ von \setcover{} sei $\chi$ die Ausgabe von $\randroundingscr$. Dann gelten + Für eine Eingabe $S$ von \setcover{} sei $\chi$ die Ausgabe von $\randroundscr$. Dann gelten \begin{enumerate} \item $P\left[\chi\,\text{ist eine Überdeckung}\right] \ge 1 - n \cdot e^{-r}$ \item $E\left[\abs{\chi}\right] \le r \cdot \opt(S)$ @@ -976,4 +978,39 @@ Damit lautet das ILP $X$ für \setcover: \end{equation*} \end{enumerate} \end{proof} + +\subsection{Las-Vegas-Algorithmus $\lasvegasscr$ für zuverlässig zulässige Lösungen} +\begin{algorithmic}[1] + \State löse das LP $X_\rel$ + \State $\tau = 0$ + \Repeat + \State $\chi = \randroundscr(S)$ + \State $\tau = \tau + 1$ + \Until{$V(\chi) = V$} + \State \Return $S_\cov = \chi$ +\end{algorithmic} +Das LP $X_\rel$ wird dabei nur einmal und nicht erneut in $\randroundscr{}$ gelöst. +$\lasvegasscr$ ist ein sogenannter Las-Vegas-Algorithmus, weil er immer eine zulässige Lösung zurückliefert, seine Laufzeit allerdings zufällig ist. + +\begin{satz} + Sei $S$ eine Eingabe von \setcover{} und gelte $r > \ln n$. Für \lasvegasscr{} gelten dann + \begin{enumerate} + \item $S_\cov$ ist eine Überdeckung mit erwarteter Größe von höchstens $r \cdot \opt(S)$. + \item Die erwartete Anzahl der Iterationen der Repeat-Schleife ist höchstens $\frac{e^{2r}}{\left(n - e^{r}\right)^2}$. + \end{enumerate} +\end{satz} +\begin{proof} + \begin{enumerate} + \item Das Ergebnis folgt direkt aus der Analyse von \randroundscr. + \item \begin{align*} + E\left[\tau\right] &= \sum_{t=1}^\infty t \cdot P\left[\text{erst die $t$. Wiederholdung ist eine Überdeckung}\right]\\ + &\le \sum_{t=1}^\infty t\cdot P\left[\text{ $t - 1$ Wiederholdungen sind keine Überdeckung}\right]\\ + &\le \sum_{t=1}^\infty t\cdot \left(n\cdot e^{-r}\right)^{t-1} \overset{\text{Konvergenz durch}\, r > \ln n}{=} \frac{e^{2r}}{\left(n - e^r\right)^2} + \end{align*} + \end{enumerate} +\end{proof} + +\begin{satz} + $\lasvegassc[\ln n + 1]$ garantiert eine relative erwartete Güte von $\ln n + 1$. Der Erwartungswert für die Anzahl der Iterationen der Schleife ist $\left(\frac{e}{e - 1}\right)^2 < 2.503$. +\end{satz} \end{document}