Regularized Non-monotone Submodular Maximization

TL;DR

This paper develops algorithms for maximizing regularized non-monotone submodular functions under various constraints, providing approximation guarantees despite the challenges posed by potential negative values of the objective.

Contribution

It introduces novel algorithms with approximation guarantees for regularized non-monotone submodular maximization, including continuous greedy and faster cardinality-constrained methods.

Findings

Continuous greedy algorithm achieves expected approximation ratio with matroid constraints.

Faster algorithm for cardinality constraints with improved query complexity.

Unconstrained maximization algorithm with near-optimal approximation ratio.

Abstract

In this paper, we present a thorough study of maximizing a regularized non-monotone submodular function subject to various constraints, i.e., $\max \{ g(A) - \ell(A) : A \in \mathcal{F} \}$, where $g \colon 2^\Omega \to \mathbb{R}_+$ is a non-monotone submodular function, $\ell \colon 2^\Omega \to \mathbb{R}_+$ is a normalized modular function and $\mathcal{F}$ is the constraint set. Though the objective function $f := g - \ell$ is still submodular, the fact that $f$ could potentially take on negative values prevents the existing methods for submodular maximization from providing a constant approximation ratio for the regularized submodular maximization problem. To overcome the obstacle, we propose several algorithms which can provide a relatively weak approximation guarantee for maximizing regularized non-monotone submodular functions. More specifically, we propose a continuous greedy algorithm for the relaxation of maximizing $g - \ell$ subject to a matroid constraint. Then, the pipage rounding procedure can produce an integral solution $S$ such that $\mathbb{E} [g(S) - \ell(S)] \geq e^{-1}g(OPT) - \ell(OPT) - O(\epsilon)$. Moreover, we present a much faster algorithm for maximizing $g - \ell$ subject to a cardinality constraint, which can output a solution $S$ with $\mathbb{E} [g(S) - \ell(S)] \geq (e^{-1} - \epsilon) g(OPT) - \ell(OPT)$ using $O(\frac{n}{\epsilon^2} \ln \frac 1\epsilon)$ value oracle queries. We also consider the unconstrained maximization problem and give an algorithm which can return a solution $S$ with $\mathbb{E} [g(S) - \ell(S)] \geq e^{-1} g(OPT) - \ell(OPT)$ using $O(n)$ value oracle queries.