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Diffstat (limited to 'main.tex')
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@@ -17,21 +17,13 @@ element $i$ to be included is the one which maximizes the \end{align} This is repeated until the sum of costs of elements in $S$ reaches the budget constraint $B$. Denote by $S_G$ the set constructed by this heuristic and let -$i^*$ be the element of maximum value. Then, the following lemma holds: -\begin{lemma}~\cite{chen}\label{lemma:greedy-bound} -Let $S_G$ be the set computed by the greedy algorithm and let -$i^* = \argmax_{i\in\mathcal{N}} V(\{i\}).$ We have: -\begin{displaymath} -OPT \leq \frac{e}{e-1}\big( 3 V(S_G) + 2 V(i^*)\big). -\end{displaymath} -\end{lemma} - -This lemma immediately implies that the following algorithm: +$i^*=\argmax_{i\in\mathcal{N}} V(\{i\})$ be the element of maximum value. Then, +the following algorithm: \begin{equation}\label{eq:max-algorithm} \textbf{if}\; V(\{i^*\}) \geq V(S_G)\; \textbf{return}\; \{i^*\} \;\textbf{else return}\; S_G \end{equation} -has an approximation ratio of $\frac{5e}{e-1}$. +has an approximation ratio of $\frac{5e}{e-1}$ \cite{chen}. \subsection{Submodular Maximization in the Strategic Case} |