On Maximizing Sums of Non-monotone Submodular and Linear Functions

TL;DR

This paper develops improved approximation algorithms and inapproximability bounds for maximizing sums of non-monotone submodular and linear functions under various constraints, advancing the theoretical understanding of these problems.

Contribution

It introduces new approximation algorithms for RegularizedUSM and RegularizedCSM, especially for the unconstrained linear case, and establishes tight inapproximability bounds.

Findings

First nontrivial approximation for RegularizedCSM with unconstrained linear functions.

Improved approximation ratio for RegularizedUSM over previous work.

Tight inapproximability bounds for certain submodular maximization problems.

Abstract

We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by Bodek and Feldman [BF22]. In this problem, you are given a non-monotone non-negative submodular function $f:2^{\mathcal N}\to \mathbb R_{\ge 0}$ and a linear function $\ell:2^{\mathcal N}\to \mathbb R$ over the same ground set $\mathcal N$, and the objective is to output a set $T\subseteq \mathcal N$ approximately maximizing the sum $f(T)+\ell(T)$. Specifically, an algorithm is said to provide an $(\alpha,\beta)$-approximation for RegularizedUSM if it outputs a set $T$ such that $\mathbb E[f(T)+\ell(T)]\ge \max_{S\subseteq \mathcal N}[\alpha \cdot f(S)+\beta\cdot \ell(S)]$. We also study the setting where $S$ and $T$ are subject to a matroid constraint, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM). For both RegularizedUSM and RegularizedCSM, we provide improved $(\alpha,\beta)$-approximation algorithms for the cases of non-positive $\ell$, non-negative $\ell$, and unconstrained $\ell$. In particular, for the case of unconstrained $\ell$, we are the first to provide nontrivial $(\alpha,\beta)$-approximations for RegularizedCSM, and the $\alpha$ we obtain for RegularizedUSM is superior to that of [BF22] for all $\beta\in (0,1)$. In addition to approximation algorithms, we provide improved inapproximability results for all of the aforementioned cases. In particular, we show that the $\alpha$ our algorithm obtains for RegularizedCSM with unconstrained $\ell$ is tight for $\beta\ge \frac{e}{e+1}$. We also show 0.478-inapproximability for maximizing a submodular function where $S$ and $T$ are subject to a cardinality constraint, improving the long-standing 0.491-inapproximability result due to Gharan and Vondrak [GV10].