From cb490cb5dc8a6f76e87b6d130c2fc6b9150c9936 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 23 Dec 2013 16:26:21 +0100
Subject: Ultimate changes before submission

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 related.tex | 5 +++++
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@@ -13,6 +13,8 @@ runs in polynomial time is presently known for submodular maximization.
 Assuming access to an oracle providing the optimum in the
 full-information setup, Chen \emph{et al.}~propose a truthful, $8.34$-approximate mechanism; in cases for which the full-information problem is NP-hard, as \EDP, this mechanism is not poly-time, unless P=NP.  Chen \emph{et al.}~also prove a $1+\sqrt{2}$ lower bound for truthful deterministic mechanisms, improving upon an earlier bound of 2 by \citeN{singer-mechanisms}. 
 
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+
 \noindent\emph{Specific Problems.}
 Improved uper and lower bounds \emph{and} deterministic polynomial mechanisms are known for specific submodular objectives \cite{singer-mechanisms, chen, singer-influence}.
 \junk{For symmetric submodular functions, a truthful mechanism with approximation ratio 2 is known, and this ratio is tight \cite{singer-mechanisms}. Singer provides a 7.32-approximate truthful mechanism for the budget feasible version of \textsc{Matching}, and a corresponding lower bound of 2 \cite{singer-mechanisms}. Improving an earlier result by Singer, \citeN{chen} give a truthful, $2+\sqrt{2}$-approximate mechanism for \textsc{Knapsack}, and a lower bound of $1+\sqrt{2}$. Finally, a truthful, 31-approximate  mechanism is also known for the budgeted version of \textsc{Coverage} \cite{singer-influence}.}
@@ -35,6 +37,8 @@ a truthful, $O(\log^3 n)$-approximate mechanism
 \cite{bei2012budget}. Moreover, in a Bayesian setup, assuming a prior distribution among the agent's costs, there exists a truthful mechanism with a 768/512-approximation ratio \cite{bei2012budget}. %(in terms of expectations)
 Posted price, rather than direct revelation mechanisms, are also studied in \cite{singerposted}.
 
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 \noindent\emph{Monotone Approximations in Combinatorial Auctions.}
 Relaxations of combinatorial problems are prevalent
 in \emph{combinatorial auctions}, 
@@ -59,6 +63,7 @@ Section~\ref{sec:monotonicity}. However, we seek a deterministic mechanism and $
 \citeN{briest-approximation} construct monotone FPTAS for problems that can be approximated through rounding techniques, which in turn can be used to construct truthful, deterministic, constant-approximation  mechanisms for corresponding  combinatorial auctions.}
 
 %\EDP{} is not readily approximable through such rounding techniques; as such, we rely on a relaxation to approximate it.
+\vspace{0.5\baselineskip}
 
 \noindent\emph{$\delta$-Truthfulness and Differential Privacy.}
 The notion of $\delta$-truthfulness has attracted considerable attention recently in the context of differential privacy (see, \emph{e.g.}, the survey by \citeN{pai2013privacy}).  \citeN{mcsherrytalwar} were the first to observe that any $\epsilon$-differentially private mechanism must also be $\delta$-truthful in expectation, for $\delta=2\epsilon$. This property was used to construct $\delta$-truthful (in expectation) mechanisms for a digital goods auction~\cite{mcsherrytalwar} and for $\alpha$-approximate equilibrium selection \cite{kearns2012}. \citeN{approximatemechanismdesign} propose a framework for converting a differentially private mechanism to a truthful-in-expectation mechanism by randomly selecting between a differentially private mechanism with good approximation guarantees, and a truthful mechanism. They apply their framework to the \textsc{FacilityLocation} problem. We depart from the above works in seeking a deterministic mechanism for \EDP, and using a stronger notion of $\delta$-truthfulness.
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