Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes

Details Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes

2011-05-23

arxiv.org/abs/1105.4593v2

Computer Science

Discrete Mathematics

Slight revision of previous version. A short version of this article appeared at STOC 2011

Scientific paper

We consider the problem of maximizing a non-negative submodular set function $f:2^N \rightarrow \mathbb{R}_+$ over a ground set $N$ subject to a variety of packing type constraints. In this paper we develop a general framework leading to a number of new results, in particular when $f$ may be a {\em non-monotone} function. Our algorithms are based on (approximately) maximizing the multilinear extension $F$ of $f$ \cite{CCPV07} over a polytope $P$ that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings \cite{CCPV09,KulikST09,LeeMNS09,CVZ10,BansalKNS10}, it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize $F$ over any down-closed polytope $P$ that has an efficient separation oracle. Previously this was known only for monotone functions \cite{Vondrak08}. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints \cite{LeeMNS09} or a matroid polytope \cite{Vondrak09,OV11}. Second, we show that {\em contention resolution schemes} are an effective way to round a fractional solution, even when $f$ is non-monotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via LP duality we show that a contention resolution scheme for a constraint is related to the {\em correlation gap} \cite{ADSY10} of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope. Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples.

Affiliated with

Also associated with

No associations

Rating

Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-23172