\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 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 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.

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}.
%
%Instead, \citeN{singer-mechanisms} and \citeN{chen}
%suggest to approximate $OPT_{-i^*}$ by a quantity $OPT'_{-i^*}$ satisfying the
%following properties:
%\begin{itemize}
%    \item $OPT'_{-i^*}$ must not depend on agent $i^*$'s reported cost and must
%    be decreasing with respect to the costs. This is to ensure the monotonicity
%    of the allocation function.
%    \item $OPT'_{-i^*}$ must be close to $OPT_{-i^*}$ to maintain a bounded
%    approximation ratio.
%    \item $OPT'_{-i^*}$ must be computable in polynomial time.
%\end{itemize}
%
%One of the main technical contributions of \citeN{chen} and
%\citeN{singer-influence} is to come up with appropriate such quantity by
%considering relaxations of \textsc{Knapsack} and \textsc{Coverage},
%respectively.
%
%\subsection{Our Approach}
%
%Using the results from Section~\ref{sec:approximation}, we come up with
%a suitable approximation $OPT_{-i^*}'$ of $OPT_{-i^*}$. Our approximation being
%$\delta$-decreasing, the resulting mechanism will be $\delta$-truthful as
%defined below.
%
%\begin{definition}
%Let $\mathcal{M}= (S, p)$ a mechanism, and let $s_i$ denote the indicator
%function of $i\in S(c)$, $s_i(c) \defeq \id_{i\in S(c)}$. We say that $\mathcal{M}$ is
%$\delta$-truthful iff:
%\begin{displaymath}
%\forall c\in\reals_+^n,\;
%\forall\mu\;\text{with}\; |\mu|\geq\delta,\;
%\forall i\in\{1,\ldots,n\},\;
%p(c)-s_i(c)\cdot c_i \geq p(c+\mu e_i) - s_i(c+\mu e_i)\cdot c_i
%\end{displaymath}
%where $e_i$ is the $i$-th canonical basis vector of $\reals^n$.
%\end{definition}
%
%Intuitively, $\delta$-truthfulness implies that agents have no incentive to lie
%by more than $\delta$ about their reported costs. In practical applications,
%the bids being discretized, if $\delta$ is smaller than the discretization
%step, the mechanism will be truthful in effect.

%$\delta$-truthfulness will follow from $\delta$-monotonicity by the following
%variant of Myerson's theorem.

\begin{lemma}\label{thm:myerson-variant}
 A normalized mechanism $\mathcal{M} = (S,p)$ for a single parameter auction is
$\delta$-truthful if:
(a) $S$ is $\delta$-monotone, \emph{i.e.}, for any agent $i$ and $c_i' \leq
c_i-\delta$, for any
fixed costs $c_{-i}$ of agents in $\mathcal{N}\setminus\{i\}$, $i\in S(c_i,
c_{-i})$ implies $i\in S(c_i', c_{-i})$, and (b)
 agents are paid \emph{threshold payments}, \emph{i.e.}, for all $i\in S(c)$, $p_i(c)=\inf\{c_i': i\in S(c_i', c_{-i})\}$.
\end{lemma}
Lemma~\ref{thm:myerson-variant} allows us to incorporate our relaxation in the above framework, yielding the following theorem:
\begin{theorem}\label{thm:main}
    For any $\delta\in(0,1]$, and any $\epsilon\in (0,1]$,  there exists a $\delta$-truthful, individually rational
    and budget feasible mechanim for \EDP{} that runs in time
    $O\big(poly(n, d, \log\log\frac{B}{b\varepsilon\delta})\big)$
    and allocates
    a set $S^*$ such that
    \begin{displaymath}
        OPT
         \leq \frac{10e-3 + \sqrt{64e^2-24e + 9}}{2(e-1)} V(S^*)+
        \varepsilon
         \simeq 12.98V(S^*) + \varepsilon.
     \end{displaymath}
\end{theorem}
The proof of the theorem, as well as our proposed mechanism, can be found in Appendix~\ref{sec:proofofmainthm}.
In addition, we prove the following simple lower bound, proved in Appendix~\ref{proofoflowerbound}.

\begin{theorem}\label{thm:lowerbound}
There is no $2$-approximate, truthful, budget feasible, individually rational
mechanism for EDP. 
\end{theorem}



