On Maximizing Sums of Non-monotone Submodular and Linear Functions

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].