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diff --git a/related.tex b/related.tex index ebc3d29..c037d86 100644 --- a/related.tex +++ b/related.tex @@ -1,5 +1,21 @@ \subsection{Related work} + +Budget feasible mechanism design was originally proposed by Singer \cite{singer-mechanism}. 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). Chen \emph{et al.}~\cite{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. + +In contrast to the above results, no 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, $8.34$-appoximate 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 $P=NP$. Moreover, Chen et al.~prove a $1+\sqrt{2}$ lower bound for truthful mechanisms, improving upon an earlier bound of 2 by Singer \cite{singer-mechanisms}. + +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, Chen \emph{et al.}~\cite{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}. + +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 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) + +\stratis{TODO: privacy discussion. Logdet objective. Should be one paragraph each.} + +\begin{comment} Two types of mechanisms: \emph{deterministic} and \emph{randomized}. For randomized mechanisms, people seek \emph{universally truthful} mechanisms: mechanisms which are a randomization of truthful mechanisms. @@ -50,3 +66,4 @@ algorithm. \thibaut{Knapsack reduces to our problem in dimension 1, hence maximizing log det is NP-hard. The approximation ratio is at least (1-1/e) by applying the general approximation algorithm from Sviridenko \cite{sviridenko-submodular}.} +\end{comment} |