Maximizing Sums of Non-monotone Submodular and Linear Functions: Understanding the Unconstrained Case

TL;DR

This paper investigates the unconstrained maximization of the sum of a non-monotone submodular function and a linear function, providing new algorithms and bounds for this general problem and its special cases.

Contribution

It introduces the first non-trivial algorithmic guarantees for the general non-monotone case and refines understanding of approximation limits for specific linear function cases.

Findings

First non-trivial guarantee for RegularizedUSM with non-monotone submodular functions.

Inapproximability results for cases with non-positive linear functions.

Improved guarantees for the Double Greedy algorithm with non-negative linear functions.

Abstract

Motivated by practical applications, recent works have considered maximization of sums of a submodular function $g$ and a linear function $\ell$. Almost all such works, to date, studied only the special case of this problem in which $g$ is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which $g$ is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function $\ell$ is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function $\ell$ is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).