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diff --git a/approximation.tex b/approximation.tex index 23dc212..670352b 100644 --- a/approximation.tex +++ b/approximation.tex @@ -1,3 +1,7 @@ +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 |