Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes

TL;DR

This paper develops a unified framework for maximizing non-negative submodular functions under complex constraints, extending approximation guarantees to non-monotone functions and combining rounding techniques for diverse polytope intersections.

Contribution

It introduces constant factor approximation algorithms for non-monotone submodular maximization over general down-closed polytopes and advances contention resolution schemes for complex constraints.

Findings

Achieved constant factor approximations for non-monotone functions over general polytopes.

Extended contention resolution schemes to handle intersections of multiple constraints.

Linked contention resolution schemes to the correlation gap, optimizing rounding methods.

Abstract

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 including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when $f$ may be a non-monotone function. Our algorithms are based on (approximately) maximizing the multilinear extension $F$ of $f$ over a polytope $P$ that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully, 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 a down-closed polytope $P$ described by an efficient separation oracle. Previously this was known only for monotone functions. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints or a matroid polytope. Second, we show that 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 correlation gap 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. Contention resolution schemes may find other applications.