diff options
Diffstat (limited to 'intro.tex')
| -rw-r--r-- | intro.tex | 2 |
1 files changed, 1 insertions, 1 deletions
@@ -64,7 +64,7 @@ We note that the objective \eqref{obj} is submodular. Using this fact, applying From a technical perspective, we propose a convex optimization problem and establish that its optimal value within a constant factor from the optimal value of \EDP. In particular, we show our relaxed objective is within a constant factor from the so-called multi-linear extension of \eqref{obj}, which in turn can be related to \eqref{obj} through pipage rounding. We establish the constant factor to the multi-linear extension by bounding the partial derivatives of these two functions; we achieve the latter by exploiting convexity properties of matrix functions over the convex cone of positive semidefinite matrices. -Our convex relaxation of \EDP{} involves maximizing a self-concordant function subject to linear constraints. Its optimal value can be computed with arbitrary accuracy in polynomial time using the so-called barrier method. However, the outcome of this computation may not nevessarily be monotone, a property needed in designing a truthful mechanism. Nevetheless, we construct an algorithm that solves the above convex relaxation and is ``almost'' monotone; in turn, we also show that this algorithm can be employed to design a $\delta$-truthful mechanism for \EDP{}. +Our convex relaxation of \EDP{} involves maximizing a self-concordant function subject to linear constraints. Its optimal value can be computed with arbitrary accuracy in polynomial time using the so-called barrier method. However, the outcome of this computation may not necessarily be monotone, a property needed in designing a truthful mechanism. Nevetheless, we construct an algorithm that solves the above convex relaxation and is ``almost'' monotone; in turn, we also show that this algorithm can be employed to design a $\delta$-truthful mechanism for \EDP{}. %This allows us to adopt the approach followed by prior work in budget feasible mechanisms by Chen \emph{et al.}~\cite{chen} and Singer~\cite{singer-influence}. %{\bf FIX the last sentence} |