Previous approaches towards designing truthful, budget feasible mechanisms for \textsc{Knapsack}~\cite{chen} and \textsc{Coverage}~\cite{singer-influence} build upon polynomial-time algorithms that compute an approximation of $OPT$, the optimal value in the full information case. Crucially, to be used in designing a truthful mechanism, such algorithms need also to be \emph{monotone}, in the sense that decreasing any cost $c_i$ leads to an increase  in the estimation of $OPT$; %In the cases of \textsc{Knapsack} and~\textsc{Coverage}, as well as in the case of \EDP{}, 
the monotonicity property  precludes using traditional approximation algorithms.

In the first part of this section, we address this issue by designing a convex relaxation of \EDP{}, and showing that its solution can be used to approximate $OPT$. The objective of this relaxation is concave and self-concordant \cite{boyd2004convex} and, as such, there exists an algorithm that solves this relaxed problem with arbitrary accuracy in polynomial time. Unfortunately, the output of this algorithm may not necessarily be monotone. Nevertheless, in the second part of this section, we show that a  solver of the relaxed problem can be used to construct a solver that is ``almost'' monotone. In Section~\ref{sec:main}, we show that this algorithm can be used to design a $\delta$-truthful mechanism for \EDP.


%As noted above, \EDP{} is NP-hard. Designing a mechanism for this problem, as
%we will see in Section~\ref{sec:mechanism}, will involve being able to find an approximation of its optimal value
%$OPT$ defined in \eqref{eq:non-strategic}. In addition to being computable in
%polynomial time and having a bounded approximation ratio to $OPT$, we will also
%require this approximation to be non-increasing in the following sense:

%\begin{definition}
%Let $f$ be a function from $\reals^n$ to $\reals$. We say that $f$ is
%\emph{non-decreasing (resp. non-increasing) along the $i$-th coordinate} iff:
%\begin{displaymath}
%\forall x\in\reals^n,\;
%t\mapsto f(x+ te_i)\; \text{is non-decreasing (resp. non-increasing)}
%\end{displaymath}
%where $e_i$ is the $i$-th canonical basis vector of $\reals^n$. 

%We say that $f$ is \emph{non-decreasing} (resp. \emph{non-increasing}) iff it
%is non-decreasing (resp. non-increasing) along all coordinates.
%\end{definition}


%Such an approximation will be obtained by introducing a concave optimization
%problem with a constant approximation ratio to \EDP{}
%(Proposition~\ref{prop:relaxation}) and then showing how to approximately solve
%this problem in a monotone way.

\subsection{A Convex Relaxation of \EDP}\label{sec:concave}

A classical way of relaxing combinatorial optimization problems is 
\emph{relaxing by expectation}, using the so-called \emph{multi-linear}
extension of the objective function $V$ (see, \emph{e.g.}, \cite{calinescu2007maximizing,vondrak2008optimal,dughmi2011convex}).
This is because this extension can yield approximation guarantees for a wide class of combinatorial problems through \emph{pipage rounding}, a technique by \citeN{pipage}.  In general, such relaxations preserve monotonicity which, as discussed, is required in mechanism design.

Formally, let $P_\mathcal{N}^\lambda$ be a probability distribution over $\mathcal{N}$ parametrized by $\lambda\in [0,1]^n$, where a set $S\subseteq \mathcal{N}$ sampled from  $P_\mathcal{N}^\lambda$ is constructed as follows:  each $i\in \mathcal{N}$ is selected to be in $S$ independently with probability $\lambda_i$, \emph{i.e.},
%\begin{displaymath}
$    P_\mathcal{N}^\lambda(S) \defeq \prod_{i\in S} \lambda_i
    \prod_{i\in\mathcal{N}\setminus S}( 1 - \lambda_i).$
%\end{displaymath}
Then, the \emph{multi-linear} extension $F:[0,1]^n\to\reals$ of $V$ is defined as the
expectation of $V$ under the  distribution $P_\mathcal{N}^\lambda$:
\begin{equation}\label{eq:multi-linear}
    F(\lambda) 
    \defeq \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[V(S)\big]
  %  = \sum_{S\subseteq\mathcal{N}} P_\mathcal{N}^\lambda(S) V(S)
= \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[ \log\det\big( I_d + \textstyle\sum_{i\in S} x_i\T{x_i}\big) \big],\quad \lambda\in[0,1]^n.
\end{equation}
Function $F$ is an extension of $V$ to the domain  $[0,1]^n$, as it equals $V$ on integer inputs: $F(\id_S) = V(S)$ for all
$S\subseteq\mathcal{N}$, where $\id_S$ denotes the indicator vector of $S$. %\citeN{pipage} have shown how to use this extension to obtain approximation guarantees for an interesting class of optimization problems through the \emph{pipage rounding} framework, which has been successfully applied in \citeN{chen, singer-influence}. 
Contrary to problems such as \textsc{Knapsack}, the
multi-linear extension \eqref{eq:multi-linear} cannot be optimized in
polynomial time for  the value function $V$  we study here, given by \eqref{modified}. Hence, we introduce an extension $L:[0,1]^n\to\reals$ s.t.~
\begin{equation}\label{eq:our-relaxation}
        L(\lambda) \defeq
\log\det\big(I_d + \textstyle\sum_{i\in\mathcal{N}} \lambda_i x_i\T{x_i}\big),
%= \log\det\left(\mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\bigg[I_d + \sum_{i\in S} x_i\T{x_i} \bigg]\right),
\quad \lambda\in[0,1]^n.
\end{equation}
Note that $L$ also extends $V$, and follows naturally from the multi-linear extension by swapping the
expectation and $\log \det$ in \eqref{eq:multi-linear}. Crucially, it is \emph{strictly concave} on $[0,1]^n$, a fact that  we exploit in the next section to maximize $L$ subject to the budget constraint in polynomial time.
%\begin{displaymath}
%    L(\lambda) = 
%\end{displaymath}

Our first technical lemma relates $L$ to the multi-linear extension $F$:
\begin{lemma}\label{lemma:relaxation-ratio}
For all $\lambda\in[0,1]^{n},$
     $           \frac{1}{2}
        \,L(\lambda)\leq
        F(\lambda)\leq L(\lambda)$.
\end{lemma}
The proof of this lemma can be found in \cite{arxiv}. In short,  exploiting the concavity of the $\log\det$ function over the set of positive semi-definite matrices, we first  bound the ratio of all partial derivatives of $F$ and $L$. We then show that the bound on the ratio of the derivatives also implies a bound on the ratio $F/L$.

Armed with this result, we subsequently use  pipage rounding to show that any $\lambda$ that maximizes the multi-linear extension $F$ can be rounded to an ``almost'' integral solution. More specifically, given a set of costs $c\in \reals^n_+$, we say that a $\lambda\in [0,1]^n$ is feasible if it belongs to the set $\dom_c =\{\lambda \in [0,1]^n: \sum_{i\in \mathcal{N}} c_i\lambda_i\leq B\}$. Then, the following lemma holds:

\begin{lemma}[Rounding]\label{lemma:rounding}
    For any feasible $\lambda\in \dom_c$, there exists a feasible
    $\bar{\lambda}\in \dom_c$ such that (a) $F(\lambda)\leq F(\bar{\lambda})$, and (b) at most one of the 
    coordinates  of $\bar{\lambda}$ is fractional. %, that is, lies in $(0,1)$ and:
\end{lemma}
The proof, also in \cite{arxiv}, follows the main steps of the pipage rounding method of \citeN{pipage}. % this  rounding property is referred to in the literature as \emph{cross-convexity} (see, \emph{e.g.}, \cite{dughmi}), or $\varepsilon$-convexity by \citeN{pipage}. 
Together, Lemma~\ref{lemma:relaxation-ratio} and Lemma~\ref{lemma:rounding} imply that $OPT$, the optimal value of \EDP, can be approximated by solving the following convex optimization problem:
\begin{align}\tag{$P_c$}\label{eq:primal}
\text{Maximize:} \quad L(\lambda)\quad \text{subject to:} \quad\lambda \in \dom_c
\end{align}
In particular, for $L_c^*\defeq \max_{\lambda\in \dom_c} L(\lambda)$, the following holds \cite{arxiv}:
\begin{proposition}\label{prop:relaxation}
$OPT\leq L^*_c \leq 2 OPT + 2\max_{i\in\mathcal{N}}V(i)$.
\end{proposition}
As we discuss in the next section, $L^*_c$ can be computed by a poly-time algorithm at arbitrary accuracy. However, the outcome of this computation may not necessarily be monotone; we address this by converting this poly-time estimator of $L^*_c$ to one that is ``almost'' monotone.%The optimization program \eqref{eq:non-strategic} extends naturally to such
%a relaxation. We define:
%\begin{equation}\tag{$P_c$}\label{eq:primal}
%    L^*_c \defeq \max_{\lambda\in[0, 1]^{n}}
%    \left\{L(\lambda) \Big| \sum_{i=1}^{n} \lambda_i c_i
%    \leq B\right\}
%\end{equation}

%The key property of our relaxation $L$ is that it has a bounded approximation
%ratio to the multi-linear relaxation $F$. This is one of our main technical
%contributions and is stated and proved in Lemma~\ref{lemma:relaxation-ratio}
%found in Appendix. Moreover, the multi-linear relaxation $F$ has an exchange
%property (see Lemma~\ref{lemma:rounding}) which allows us to relate its value to
%$OPT$ through rounding. Together, these properties give the following
%proposition which is also proved in the Appendix.

\subsection{Polynomial-Time, Almost-Monotone Approximation}\label{sec:monotonicity}
The $\log\det$ objective function of \eqref{eq:primal} is strictly concave and \emph{self-concordant}~\cite{boyd2004convex}. The maximization of a concave, self-concordant function subject to a set of linear constraints can be performed through the \emph{barrier method} (see, \emph{e.g.},~\cite{boyd2004convex} Section 11.5.5 for  general self-concordant optimization as well as \cite{vandenberghe1998determinant} for a detailed treatment of the $\log\det$ objective). The performance of the barrier method is summarized in our case by the following lemma:
\begin{lemma}[\citeN{boyd2004convex}]\label{lemma:barrier}
For any $\varepsilon>0$, the barrier method computes  an 
approximation $\hat{L}^*_c$ that is $\varepsilon$-accurate, \emph{i.e.}, it satisfies $|\hat L^*_c- L^*_c|\leq \varepsilon$, in time $O\left(poly(n,d,\log\log\varepsilon^{-1})\right)$. The same guarantees apply when maximizing $L$ subject to an arbitrary set of $O(n)$ linear constraints.
\end{lemma}

 Clearly, the optimal value $L^*_c$ of \eqref{eq:primal} is monotone in $c$: formally, for any two $c,c'\in \reals_+^n$ s.t.~$c\leq c'$ coordinate-wise, $\dom_{c'}\subseteq \dom_c$ and thus $L^*_c\geq L^*_{c'}$. Hence, the map $c\mapsto L^*_c$ is non-increasing. Unfortunately, the same is not true for the output $\hat{L}_c^*$ of the barrier method: there is no guarantee that the $\epsilon$-accurate approximation $\hat{L}^*_c$ exhibits any kind of monotonicity.

Nevertheless, we prove that it is possible to use the barrier method to construct an approximation of $L^*_{c}$ that is ``almost'' monotone. More specifically, given $\delta>0$,  we say that $f:\reals^n\to\reals$ is
\emph{$\delta$-decreasing}  if
%\begin{equation}\label{eq:dd}
 $ f(x) \geq  f(x+\mu e_i)$, 
    for all $i\in \mathcal{N},x\in\reals^n, \mu\geq\delta,$
%\end{equation}
where $e_i$ is the $i$-th canonical basis vector of $\reals^n$.
In other words, $f$ is $\delta$-decreasing if increasing any coordinate by $\delta$ or more at input $x$ ensures that the output will be at most $f(x)$.

%Our next technical result establishes that, using the barrier method, it is possible to construct an algorithm that computes $L^*_c$ at arbitrary accuracy in polynomial time and is $\delta$-decreasing.

We achieve this by restricting the optimization over a subset of $\dom_c$ at which the concave relaxation $L$ is ``sufficiently'' concave. Formally, for $\alpha\geq 0$ let
$$\textstyle\dom_{c,\alpha} \defeq \{\lambda \in [\alpha,1]^n: \sum_{i\in \mathcal{N}}c_i\lambda_i \leq B\}\subseteq  \dom_c . $$ 
Note that $\dom_c=\dom_{c,0}.$ Consider the following perturbed problem:
\begin{align}\tag{$P_{c,\alpha}$}\label{eq:perturbed-primal}
\begin{split}  \text{Maximize:} &\qquad L(\lambda)\\
\text{subject to:} & \qquad\lambda \in \dom_{c,\alpha}
\end{split}
\end{align}

The restricted set $\dom_{c,\alpha}$ ensures that the partial derivatives of the optimal solution to $P_{c,\alpha}$ with respect to the costs are bounded from below. This implies that an approximate solution to $P_{c,\alpha}$ given by the barrier method is $\delta$-decreasing with respect to the costs. On the other hand, by taking $\alpha$ small enough, we ensure that the approximate solution to $P_{c,\alpha}$ is still an $\epsilon$-accurate approximation of $L_c^*$. This methodology is summarized in the following proposition whose proof can be found in \cite{arxiv}.

\begin{proposition}\label{prop:monotonicity}
    For any $\delta\in(0,1]$ and any $\varepsilon\in(0,1]$, using the barrier
    method to solve \eqref{eq:perturbed-primal} for $\alpha\defeq\varepsilon
    (\delta/B+n^2)^{-1}$ with accuracy $\frac{1}{2^{n+1}B}\alpha\delta b$
    yields a $\delta$-decreasing, $\varepsilon$-accurate approximation of
    $L^*_c$. The running time of the algorithm is $O\big(poly(n, d,
    \log\log\frac{B}{b\varepsilon\delta})\big)$.
\end{proposition}
The proof can be found in \cite{arxiv}.
