Symmetry and approximability of submodular maximization problems

TL;DR

This paper introduces a general method using symmetry gaps to establish inapproximability results for submodular maximization in the value oracle model, unifying and extending known hardness results.

Contribution

It presents a unified framework for deriving inapproximability results based on symmetry gaps, and applies it to show hardness for maximizing submodular functions over matroid bases.

Findings

No constant-factor approximation exists for maximizing non-negative submodular functions over matroid bases.

The paper provides a matching approximation algorithm for this problem.

The approach unifies several existing hardness results and introduces new inapproximability bounds.

Abstract

A number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem. These results hold typically in the value oracle model, where the objective function is accessible via a black box returning f(S) for a given S. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of symmetry gap. Our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, and implies several new ones. In particular, we prove that there is no constant-factor approximation for the problem of maximizing a non-negative submodular function over the bases of a matroid. We also provide a closely matching approximation algorithm for this problem.