From 755c7cc3dd07c3b081be23966f55a0066cdd4271 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Wed, 3 Jul 2013 09:21:24 -0700
Subject: approx

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 approximation.tex | 4 ++++
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+Previous approaches towards designing truthful, budget feasible mechanisms for \textsc{Knapsack}~\cite{chen} and the \textsc{Coverage}~\cite{singer-influence} build upon polynomial-time algorithms that compute an approximation of $OPT$, the optimal value in the full information case. Crucially, to be used in designing a truthful mechanism, such algorithms need also to be \emph{monotone}, in the sense that decreasing any of the costs $c_i$ leads to an increase of the estimate of $OPT$. Both in the cases of \textsc{Knapsack}~\cite{chen} and~\textsc{Coverage}, as well as in the case of \EDP{}, the monotonicity requirement prevents using traditional approximation algorithms for approximating any of the above problems.
+
+We address this issue by designing a convex relaxation of \EDP{}, and showing that it well approximates $OPT$. 
+
 As noted above, \EDP{} is NP-hard. Designing a mechanism for this problem, as
 we will see in Section~\ref{sec:mechanism}, will involve being able to find an approximation of its optimal value
 $OPT$ defined in \eqref{eq:non-strategic}. In addition to being computable in
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