From 62c98a8cc5c9f3feb18ccc4ccc2c5d11d850c020 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Sat, 6 Jul 2013 13:47:50 -0700
Subject: linear constraints

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

(limited to 'intro.tex')

diff --git a/intro.tex b/intro.tex
index 3a2925b..4d8b5dd 100644
--- a/intro.tex
+++ b/intro.tex
@@ -64,7 +64,7 @@ We note that the objective \eqref{obj} is submodular. Using this fact, applying
 From a technical perspective, we propose a convex optimization problem and establish that its optimal value within a constant factor from the optimal value of \EDP.  
  In particular, we show our relaxed objective is within a constant factor from the so-called multi-linear extension of  \eqref{obj}, which in turn can be related to \eqref{obj} through pipage rounding. We establish the constant factor to the multi-linear extension by bounding  the partial derivatives of these two functions; we achieve the latter by exploiting convexity properties of matrix functions over the convex cone of positive semidefinite matrices.
 
-Our convex relaxation of \EDP{} involves maximizing a self-concordant function subject to linear constraints. Its optimal value can be computed with arbitrary accuracy in polynomial time using the so-called barrier method. However, the outcome of this computation may not nevessarily be monotone, a property needed in designing a truthful mechanism. Nevetheless, we construct an algorithm that solves the above convex relaxation and is ``almost'' monotone; in turn, we also show that this algorithm can be employed to design a $\delta$-truthful mechanism for \EDP{}.
+Our convex relaxation of \EDP{} involves maximizing a self-concordant function subject to linear constraints. Its optimal value can be computed with arbitrary accuracy in polynomial time using the so-called barrier method. However, the outcome of this computation may not necessarily be monotone, a property needed in designing a truthful mechanism. Nevetheless, we construct an algorithm that solves the above convex relaxation and is ``almost'' monotone; in turn, we also show that this algorithm can be employed to design a $\delta$-truthful mechanism for \EDP{}.
 
 
 %This allows us to adopt the approach followed by prior work in budget feasible mechanisms by Chen \emph{et al.}~\cite{chen} and Singer~\cite{singer-influence}.   %{\bf FIX the last sentence}
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