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@@ -1,13 +1,13 @@ \label{sec:main} -In this section we use the $\delta$-decreasing, $\epsilon$-accurate algorithm solving the convex optimization problem \eqref{eq:primal} to design a mechanism for \SEDP. The construction follows a methodology employed by \citeN{chen} and \citeN{singer-influence} to construct deterministic, truthful mechanisms for \textsc{Knapsack} and \textsc{Coverage} respectively. We briefly outline this below. +In this section we use the $\delta$-decreasing, $\epsilon$-accurate algorithm solving the convex optimization problem \eqref{eq:primal} to design a mechanism for \SEDP. The construction follows a methodology proposed in \cite{singer-mechanisms} and employed by \citeN{chen} and \citeN{singer-influence} to construct deterministic, truthful mechanisms for \textsc{Knapsack} and \textsc{Coverage} respectively. We briefly outline this below (see also Algorithm~\ref{mechanism} in Appendix~\ref{sec:proofofmainthm} for a detailed description). %In the strategic case, Algorithm~\eqref{eq:max-algorithm} breaks incentive compatibility. Indeed, \citeN{singer-influence} notes that this allocation function is not monotone, which implies by Myerson's theorem (Theorem~\ref{thm:myerson}) that the resulting mechanism is not truthful. -Recall from Section~\ref{sec:fullinfo} that $i^*\defeq \arg\max_{i\in \mathcal{N}} V(\{i\})$ is the element of maximum value, and $S_G$ is a set constructed greedily, by selecting elements of maximum marginal value per cost. The general framework used by \citeN{chen} and by \citeN{singer-influence} for the \textsc{Knapsack} and \textsc{Coverage} value functions contructs an allocation as follows. First, a polynomial-time, monotone relaxation of $OPT$ is computed over all elements excluding $i^*$. The outcome of this relaxation is compared to $V(\{i^*\})$: if it exceeds $V(\{i^*\})$, then the mechanism constructs an allocation $S_G$ greedily; otherwise, the only item allocated is the singleton $\{i^*\}$. +Recall from Section~\ref{sec:fullinfo} that $i^*\defeq \arg\max_{i\in \mathcal{N}} V(\{i\})$ is the element of maximum value, and $S_G$ is a set constructed greedily, by selecting elements of maximum marginal value per cost. The general framework used by \citeN{chen} and by \citeN{singer-influence} for the \textsc{Knapsack} and \textsc{Coverage} value functions contructs an allocation as follows. First, a polynomial-time, monotone approximation of $OPT$ is computed over all elements excluding $i^*$. The outcome of this approximation is compared to $V(\{i^*\})$: if it exceeds $V(\{i^*\})$, then the mechanism constructs an allocation $S_G$ greedily; otherwise, the only item allocated is the singleton $\{i^*\}$. Provided that the approximation used is within a constant from $OPT$, the above allocation can be shown to also yield a constant approximation to $OPT$. Furthermore, this allocation combined with \emph{threshold payments} (see Lemma~\ref{thm:myerson-variant} below), the mechanism can be shown to be truthful using Myerson's Theorem~\cite{myerson}. -Provided that the relaxation used within a constant from $OPT$, the above allocation can be shown to also yield a constant approximation to $OPT$. Furthermore, this allocation combined with \emph{threshold payments}, the mechanism can be shown to be truthful when the relaxation is monotone. The relaxations used in \cite{chen,singer-influence} are linear programs, and thus their optimal values are monotone and can be computed exactly in weakly polynomial time. In contrast, we extend these results by showing that the convex relaxation \eqref{eq:primal}, which can be solved by an $\epsilon$-accurate, $\delta$-decreasing algorithm, can be used to construct a $\delta$-truthful, constant approximation mechanism. +The approximation algorithms used in \cite{chen,singer-influence} are LP relaxations, and thus their outputs are monotone and can be computed exactly in polynomial time. We show that the convex relaxation \eqref{eq:primal}, which can be solved by an $\epsilon$-accurate, $\delta$-decreasing algorithm, can be used to construct a $\delta$-truthful, constant approximation mechanism, by being incorporated in the same framework. To obtain this result, we use the following modified version of Myerson's theorem \cite{myerson}, whose proof we provide in Appendix~\ref{sec:myerson}. % |