Constrained Submodular Maximization via New Bounds for DR-Submodular Functions

TL;DR

This paper introduces a new solver for constrained DR-submodular maximization that achieves a 0.401-approximation, narrowing the gap towards the theoretical limit, by developing a novel bound for DR-submodular functions.

Contribution

The authors present a new approximation algorithm for DR-submodular maximization under constraints, based on a novel bound that improves previous bounds and enhances the solver's performance.

Findings

Achieves a 0.401-approximation guarantee.

Introduces a new bound for DR-submodular functions.

Potentially applicable to a wide range of related problems.

Abstract

Submodular maximization under various constraints is a fundamental problem studied continuously, in both computer science and operations research, since the late $1970$'s. A central technique in this field is to approximately optimize the multilinear extension of the submodular objective, and then round the solution. The use of this technique requires a solver able to approximately maximize multilinear extensions. Following a long line of work, Buchbinder and Feldman (2019) described such a solver guaranteeing $0.385$-approximation for down-closed constraints, while Oveis Gharan and Vondr\'ak (2011) showed that no solver can guarantee better than $0.478$-approximation. In this paper, we present a solver guaranteeing $0.401$-approximation, which significantly reduces the gap between the best known solver and the inapproximability result. The design and analysis of our solver are based on a novel bound that we prove for DR-submodular functions. This bound improves over a previous bound due to Feldman et al. (2011) that is used by essentially all state-of-the-art results for constrained maximization of general submodular/DR-submodular functions. Hence, we believe that our new bound is likely to find many additional applications in related problems, and to be a key component for further improvement.