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@@ -5,7 +5,7 @@ In this section we use the $\delta$-decreasing, $\epsilon$-accurate algorithm so %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 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}. +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, using Myerson's Theorem~\cite{myerson}, it can be shown that this allocation combined with \emph{threshold payments} (see Lemma~\ref{thm:myerson-variant} below) constitute a truthful 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. |