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Diffstat (limited to 'proofs.tex')
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1 files changed, 17 insertions, 3 deletions
@@ -32,13 +32,27 @@ The complexity of the mechanism is given by the following lemma. \end{proof} Finally, we prove the approximation ratio of the mechanism. We use the -following lemma which establishes that $OPT'$, the optimal value \eqref{relax} of the fractional relaxation $L$ under the budget constraints - is not too far from $OPT$. +following lemma which establishes that $OPT'$, the optimal value \eqref{relax} +of the fractional relaxation $L$ under the budget constraints is not too far +from $OPT$. \begin{lemma}[Approximation]\label{lemma:relaxation} $ OPT' \leq 2 OPT + 2\max_{i\in\mathcal{N}}V(i)$. \end{lemma} -The proof of Lemma~\ref{lemma:relaxation} is our main technical contribution, and can be found in Section \ref{sec:relaxation}. +The proof of Lemma~\ref{lemma:relaxation} is our main technical contribution, +and can be found in Section \ref{sec:relaxation}. + +We also use the following lemma from \cite{chen} which bounds $OPT$ in terms of +the value of $S_G$, as computed in Algorithm \ref{mechanism}, and $i^*$, the +element of maximum value. + +\begin{lemma}[\cite{chen}] +Let $S_G$ be the set computed in Algorithm \ref{mechanism} 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} Using Lemma~\ref{lemma:relaxation} we can complete the proof of Theorem~\ref{thm:main} by showing that, for any $\varepsilon > 0$, if |