\label{sec:main}

In this section we present our mechanism for \SEDP. Prior approaches to budget
feasible mechanisms for submodular maximization build upon the full information
case, which we discuss first.

\subsection{Submodular Maximization in the Full Information Case}

In the full-information case, submodular maximization under a budget constraint
\eqref{eq:non-strategic} relies on a greedy heuristic in which elements are
added to the solution set according to the following greedy selection rule.
Assume that $S\subseteq\mathcal{N}$ is the set constructed thus far, the next
element $i$ to be included is the one which maximizes the
\emph{marginal-value-per-cost}:
\begin{align}
    i = \argmax_{j\in\mathcal{N}\setminus S}\frac{V(S\cup\{i\}) - V(S)}{c_i}\label{greedy}
\end{align}
This is repeated until the sum of costs of elements in $S$ reaches the budget
constraint $B$. Denote by $S_G$ the set constructed by this heuristic and let
$i^*=\argmax_{i\in\mathcal{N}} V(\{i\})$ be the element of maximum value. Then,
the following algorithm:
\begin{equation}\label{eq:max-algorithm}
\textbf{if}\; V(\{i^*\}) \geq V(S_G)\; \textbf{return}\; \{i^*\}
\;\textbf{else return}\; S_G
\end{equation}
has a constant approximation ratio \cite{singer-mechanisms}.

\subsection{Submodular Maximization in the Strategic Case}

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.

Let us denote by $OPT_{-i^*}$ the optimal value achievable in the
full-information case after removing $i^*$ from the set $\mathcal{N}$:
\begin{equation}\label{eq:opt-reduced}
    OPT_{-i^*} \defeq \max_{S\subseteq\mathcal{N}\setminus\{i^*\}} \Big\{V(S) \;\Big| \;
    \sum_{i\in S}c_i\leq B\Big\}
\end{equation}
\citeN{singer-mechanisms} and \citeN{chen} prove that the following
allocation:
\begin{displaymath}
\textbf{if}\; V(\{i^*\}) \geq C \cdot OPT_{-i^*}\; \textbf{return}\; \{i^*\}
\;\textbf{else return}\; S_G
\end{displaymath}
yields a 8.34-approximation mechanism for an appropriate $C$ (see also
Algorithm~\ref{mechanism}). However, $OPT_{-i^*}$, the maximum value attainable
in the full-information case, cannot be computed in poly-time when the
underlying problem is NP-hard (unless P=NP), as is the case for \SEDP{}.

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}
\sloppy A normalized mechanism $\mathcal{M} = (S,p)$ for a single parameter auction is
$\delta$-truthful iff:
(a) $S$ is $\delta$-decreasing, \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}

\begin{algorithm}[t]
    \caption{Mechanism for \SEDP{}}\label{mechanism}
    \begin{algorithmic}[1]
    \State $\mathcal{N} \gets \mathcal{N}\setminus\{i\in\mathcal{N} : c_i > B\}$
    \State $i^* \gets \argmax_{j\in\mathcal{N}}V(j)$
    \State $OPT'_{-i^*} \gets$ using Proposition~\ref{prop:monotonicity},
    compute a $\varepsilon$-accurate, $\delta$-decreasing
    approximation of $\max_{\lambda\in[0,1]^{n}} \{L(\lambda)
                                    \mid \lambda_{i^*}=0,\sum_{i \in \mathcal{N}\setminus\{i^*\}}c_i\lambda_i\leq B\}$
        \Statex
        \If{$OPT'_{-i^*} < C \cdot V(i^*)$} \label{c}
            \State \textbf{return} $\{i^*\}$ 
        \Else
            \State $i \gets \argmax_{1\leq j\leq n}\frac{V(j)}{c_j}$
            \State $S_G \gets \emptyset$
            \While{$c_i\leq \frac{B}{2}\frac{V(S_G\cup\{i\})-V(S_G)}{V(S_G\cup\{i\})}$}
                \State $S_G \gets S_G\cup\{i\}$
                \State $i \gets \argmax_{j\in\mathcal{N}\setminus S_G}
                \frac{V(S_G\cup\{j\})-V(S_G)}{c_j}$
            \EndWhile
            \State \textbf{return} $S_G$
        \EndIf
    \end{algorithmic}
\end{algorithm}

\fussy
Our mechanism for \EDP{} is composed of
\begin{itemize}
\item the allocation function presented in Algorithm~\ref{mechanism}, and 
\item the payment function which pays each allocated agent $i$ her threshold
payment as described in Myerson's Theorem.  In the case where $\{i^*\}$ is the
allocated set, her threshold payment is $B$ (she would be have been dropped on
line 1 of Algorithm~\ref{mechanism} had she reported a higher cost).
A closed-form formula for threshold payments when $S_G$ is the allocated set can
be found in~\cite{singer-mechanisms}.
\end{itemize}

We can now state our main result:
\begin{theorem}\label{thm:main}
    \sloppy
    For any $\delta$, the allocation described in Algorithm~\ref{mechanism},
    along with threshold payments, is $\delta$-truthful, individually rational
    and budget feasible.  Furthermore, there exists an absolute constant $C$
    such that, for any $\varepsilon>0$, the mechanism runs in time
    $O\big(poly(n, d, \log\log\frac{1}{b\varepsilon\delta})\big)$
    and returns
    a set $S^*$ such that:
    \begin{align*}
        OPT
        & \leq \frac{10e-3 + \sqrt{64e^2-24e + 9}}{2(e-1)} V(S^*)+
        \varepsilon
         \simeq 12.98V(S^*) + \varepsilon
    \end{align*}
\end{theorem}
The value of the constant $C$ is given by \eqref{eq:constant} in
Section~\ref{sec:proofofmainthm}.
In addition, we prove the following simple lower bound.

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

