From 51e54c074df56a4657012c42628125cc0c7a3619 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Sun, 22 Sep 2013 16:13:45 -0400
Subject: Reduce intro+related

---
 related.tex | 11 +++++------
 1 file changed, 5 insertions(+), 6 deletions(-)

(limited to 'related.tex')

diff --git a/related.tex b/related.tex
index 8cca712..c3d4ed6 100644
--- a/related.tex
+++ b/related.tex
@@ -3,7 +3,7 @@
 
 \junk{\subsection{Experimental Design} The classic experimental design problem, which we also briefly review in Section~\ref{sec:edprelim}, deals with  which $k$ experiments to conduct among a set of $n$ possible experiments. It is a well studied problem both in the non-Bayesian \cite{pukelsheim2006optimal,atkinson2007optimum,boyd2004convex} and Bayesian setting \cite{chaloner1995bayesian}. Beyond $D$-optimality, several other objectives are encountered in the literature  \cite{pukelsheim2006optimal}; many involve some function of the covariance matrix of the estimate of $\beta$, such as $E$-optimality (maximizing the smallest eigenvalue of the covariance of $\beta$) or $T$-optimality (maximizing the trace). Our focus on $D$-optimality is motivated by both its tractability as well as its relationship to the information gain.  %are encountered in the literature, though they do not relate to entropy as $D$-optimality. We leave the task of approaching the maximization of such objectives from a strategic point of view as an open problem.
 }
-\paragraph{Budget Feasible Mechanisms for General Submodular Functions}
+\noindent\emph{Budget Feasible Mechanisms for General Submodular Functions}
 Budget feasible mechanism design was originally proposed by  \citeN{singer-mechanisms}. Singer considers the problem of maximizing an arbitrary submodular function subject to a budget constraint in the \emph{value query} model, \emph{i.e.} assuming an  oracle providing the  value of the submodular objective on any given set. 
  Singer shows that there exists a randomized, 112-approximation mechanism for submodular maximization that is \emph{universally truthful} (\emph{i.e.}, it is a randomized mechanism sampled from a distribution over truthful mechanisms). \citeN{chen}  improve this result by providing a 7.91-approximate mechanism, and show a corresponding lower bound of $2$ among universally truthful randomized mechanisms for submodular maximization.
 
@@ -16,7 +16,7 @@ However, 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 the one we consider here, 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}. 
 
 
-\paragraph{Budget Feasible Mechanism Design on Specific Problems}
+\noindent\emph{Budget Feasible Mechanism Design on Specific Problems}
 Improved bounds, as well as deterministic polynomial mechanisms, are known for specific submodular objectives. For symmetric submodular functions, a truthful mechanism with approximation ratio 2 is known, and this ratio is tight \cite{singer-mechanisms}. Singer also 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}. 
 
 The  deterministic mechanisms for \textsc{Knapsack} \cite{chen} and
@@ -33,14 +33,14 @@ establish that it can be incorporated in the framework of
 
 %Our results therefore add \SEDP{} to the set of problems for which a deterministic, polynomial time, constant approximation mechanism is known.
 
-\paragraph{Beyond Submodular Objectives}
+\noindent\emph{Beyond Submodular Objectives}
 Beyond submodular objectives, it is known that no truthful mechanism with approximation ratio smaller than $n^{1/2-\epsilon}$ exists for maximizing  fractionally subadditive functions (a class that includes submodular functions) assuming access to a value query oracle~\cite{singer-mechanisms}. Assuming access to a stronger oracle (the \emph{demand} oracle), there exists
 a truthful, $O(\log^3 n)$-approximate mechanism 
 \cite{dobz2011-mechanisms} as well as a universally truthful, $O(\frac{\log n}{\log \log n})$-appro\-xi\-mate mechanism for subadditive maximization 
 \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}.
 
-\paragraph{Monotone Approximations in Combinatorial Auctions}
+\noindent\emph{Monotone Approximations in Combinatorial Auctions}
 Relaxations of combinatorial problems are prevalent
 in \emph{combinatorial auctions}, % \cite{archer-approximate,lavi-truthful,dughmi-truthful,briest-approximation}, 
 in which an auctioneer aims at maximizing a set function which is the sum of utilities of strategic bidders (\emph{i.e.}, the social welfare). As noted by \citeN{archer-approximate},
@@ -62,8 +62,7 @@ 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.
 
-\paragraph{$\delta$-Truthfulness and Differential Privacy}
-
+\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.
 
 
-- 
cgit v1.2.3-70-g09d2

