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diff --git a/related.tex b/related.tex index ec22041..9573309 100644 --- a/related.tex +++ b/related.tex @@ -6,19 +6,19 @@ Budget feasible mechanism design was originally proposed by Singer \cite{singer- 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. Chen \emph{et al.}~also 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}. +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{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) - A series of recent papers \cite{mcsherrytalwar,approximatemechanismdesign,xiao:privacy-truthfulness,chen:privacy-truthfulness} consider the related problem of conducting retreive data from an \textit{unverified} database: the buyer of the data cannot verify the data reported by individuals and therefore must incentivize them to report truthfully. -McSherry and Talwar \cite{mcsherrytalwar} argue that \emph{differentially private} mechanisms also offer a form of \emph{approximate truthfulness}, as the mechanism's output is only slightly purturbed when an individual unilaterally changes her data value; as a result, reporting untruthfully can only increase an individual's utility by a small amount. Xiao \cite{xiao:privacy-truthfulness}, improving upon earlier work by Nissim et al.~\cite{approximatemechanismdesign} construct mechanisms that + A series of recent papers \cite{mcsherrytalwar,approximatemechanismdesign,xiao:privacy-truthfulness,chen:privacy-truthfulness} consider the related problem of conducting retrieve data from an \textit{unverified} database: the buyer of the data cannot verify the data reported by individuals and therefore must incentivize them to report truthfully. +McSherry and Talwar \cite{mcsherrytalwar} argue that \emph{differentially private} mechanisms also offer a form of \emph{approximate truthfulness}, as the mechanism's output is only slightly perturbed when an individual unilaterally changes her data value; as a result, reporting untruthfully can only increase an individual's utility by a small amount. Xiao \cite{xiao:privacy-truthfulness}, improving upon earlier work by Nissim \emph{et al.}~\cite{approximatemechanismdesign} construct mechanisms that simultaneously achieve exact truthfulness as well as differential privacy. Eliciting private data through a \emph{survey} \cite{roth-liggett}, whereby individuals first decide whether to participate in the survey and then report their data, also fall under the unverified database setting \cite{xiao:privacy-truthfulness}. Our work differs from the above works in that it does not capture user utility through differential privacy; crucially, we also assume that experiments conducted are \emph{tamper-proof}, and individuals can missreport their costs but not their values. -Our work is closest to Roth and Schoenebeck \cite{roth-shoenebeck}, who also consider how to sample individuals from a population to obtain an unbiased estimate of a reported value. The authors assume a prior on the joint distribution +Our work is closest to Roth and Schoenebeck \cite{roth-schoenebeck}, who also consider how to sample individuals from a population to obtain an unbiased estimate of a reported value. The authors assume a prior on the joint distribution \stratis{to be continued} |