From 755c7cc3dd07c3b081be23966f55a0066cdd4271 Mon Sep 17 00:00:00 2001 From: Stratis Ioannidis Date: Wed, 3 Jul 2013 09:21:24 -0700 Subject: approx --- approximation.tex | 4 ++++ 1 file changed, 4 insertions(+) (limited to 'approximation.tex') 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 -- cgit v1.2.3-70-g09d2