From 8562564268f83ca4ea0a4d6cd316a61f8b66f6b2 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Sun, 7 Jul 2013 21:21:23 -0700
Subject: minor

---
 approximation.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index 6cc9e49..b1d3453 100644
--- a/approximation.tex
+++ b/approximation.tex
@@ -1,7 +1,7 @@
 Previous approaches towards designing truthful, budget feasible mechanisms for \textsc{Knapsack}~\cite{chen} and \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 cost $c_i$ leads to an increase  in the estimation of $OPT$; %In the cases of \textsc{Knapsack} and~\textsc{Coverage}, as well as in the case of \EDP{}, 
 the monotonicity property  precludes using traditional approximation algorithms.
 
-In the fist part of this section, we address this issue by designing a convex relaxation of \EDP{}, and showing that its solution can be used to approximate $OPT$. The objective of this relaxation is concave and self-concordant \cite{boyd2004convex} and, as such, there exists an algorithm that solves this relaxed problem with arbitrary accuracy in polynomial time. Unfortunately, the output of this algorithm may not necessarily be monotone. Nevertheless, in the second part of this section, we show how a  solver of the relaxed problem can be used to construct a solver that is ``almost'' monotone: in Section~\ref{sec:main}, we show how this algorithm can be used to design a $\delta$-truthful mechanism for \EDP.
+In the fist part of this section, we address this issue by designing a convex relaxation of \EDP{}, and showing that its solution can be used to approximate $OPT$. The objective of this relaxation is concave and self-concordant \cite{boyd2004convex} and, as such, there exists an algorithm that solves this relaxed problem with arbitrary accuracy in polynomial time. Unfortunately, the output of this algorithm may not necessarily be monotone. Nevertheless, in the second part of this section, we show that a  solver of the relaxed problem can be used to construct a solver that is ``almost'' monotone. In Section~\ref{sec:main}, we show that this algorithm can be used to design a $\delta$-truthful mechanism for \EDP.
 
 
 %As noted above, \EDP{} is NP-hard. Designing a mechanism for this problem, as
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