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+\subsection{Deterministisches Runden mit \detroundsc} +\begin{algorithmic}[1] + \State $(x_1, \dots, x_m) = $ löse $X_\rel$ + \State $S_\cov = \emptyset$ + \For{$I = 1\dots m$} + \If{$x_i \ge \frac{1}{\Delta_S}$} + \State $S_\cov = S_\cov \cup \{S_i\}$ + \EndIf + \EndFor + \State \Return $S_\cov$ +\end{algorithmic} + +\begin{satz} + \detroundsc{} berechnet für eine Instanz $S$ von \setcover{} in Zeit $\bigO(m + L(nm, m))$ eine Überdeckung. + Ferner gilt $\detroundsc(S) \le \Delta_S \cdot \opt(S)$. +\end{satz} +\begin{proof} + Zu einem beliebigen Objekt $u\in V$ gehört die Nebenbedingung $\sum_{i: u \in S_i} x_i \ge 1$ mit $\degree_S(u)$ Summanden. + Mit einem Durchschnittsargument kann jetzt also gefolgert werden, dass für einen der Summanden $x_i$ gilt: + \begin{equation*} + x_i \ge \frac{1}{\degree_S(u)} \ge \frac{1}{\Delta_S} + \end{equation*} + Damit wird für jedes $u$ mindestens ein überdeckendes $S_i$ aufgenommen, also stellt die Ausgabe eine Überdeckung dar. + + Weil für jede Gruppe $S_i \in S_\cov$ $\Delta_s \cdot x_i \ge 1$ gilt, folgt für die Qualität der Ausgabe + \begin{equation*} + \detroundsc(S) = \abs{S_\cov} \le \sum_{i=1}^{m} \Delta_s \cdot x_i =\Delta_S \cdot \sum_{i=1}^{m} x_i = \Delta_S \cdot \opt(X_\rel) \le \Delta_S \cdot \opt(S) + \end{equation*} +\end{proof} \end{document}