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diff --git a/related.tex b/related.tex index 4c17d1f..60bc295 100755 --- a/related.tex +++ b/related.tex @@ -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}. +\vspace{0.5\baselineskip} + \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}. +\vspace{0.5\baselineskip} + \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. |