From dd7111bf1fd35d39f5a1e4fe930cb093888fd31d Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Mon, 11 Feb 2013 14:32:13 -0800
Subject: related

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 \subsection{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 mechanisms for submodular maximization.
+ 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.
 
-In contrast to the above results, no deterministic, truthful, constant approximation mechanism that runs in polynomial time is presently known for submodular maximization.  However, assuming access to an oracle providing the optimum in the full-information setup, Chen \emph{et al.},~provide 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 mechanisms, improving upon an earlier bound of 2 by \citeN{singer-mechanisms}. 
+The above approximation guarantees hold \emph{in expectation}: for a given instance, the approximation ratio provided by the mechanism may in fact be unbounded. No deterministic, truthful, constant approximation mechanism that runs in polynomial time is presently known for submodular maximization.  However, assuming access to an oracle providing the optimum in the full-information setup, Chen \emph{et al.},~provide 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}. 
 
 
 \subsection{Budget Feasible Mechanism Design on Specific Problems}
-Improved bounds, as well as deterministic polynomial mechanisms, are known for specific submodular objectives under the value query model. 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-mechanisms,singer-influence}. Our results therefore extend the set of problems for which a deterministic, polynomial mechanism is known to include \SEDP.
+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}. Our results therefore add \SEDP{} to the set of problems for which a deterministic, polynomial time, constant approximation mechanism is known.
 
 \subsection{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, $O(\frac{\log n}{\log \log n})$-approximate 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}.
 
 
 \subsection{Data Markets}
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