Stronger Approximation Guarantees for Non-Monotone {\gamma}-Weakly DR-Submodular Maximization

TL;DR

This paper introduces an approximation algorithm for maximizing non-monotone $ ext{γ}$-weakly DR-submodular functions under constraints, providing improved guarantees that smoothly depend on $ ext{γ}$ and generalize previous bounds.

Contribution

The paper presents a novel approximation algorithm combining Frank-Wolfe and double-greedy methods, achieving state-of-the-art guarantees for non-monotone $ ext{γ}$-weakly DR-submodular maximization.

Findings

Guarantees depend smoothly on $ ext{γ}$, recovering 0.401 for $ ext{γ}=1$

Improves upon previous bounds for $ ext{γ}$-weakly DR-submodular maximization

Effective handling of non-monotonicity with a simple procedure

Abstract

Maximizing submodular objectives under constraints is a fundamental problem in machine learning and optimization. We study the maximization of a nonnegative, non-monotone $\gamma$-weakly DR-submodular function over a down-closed convex body. Our main result is an approximation algorithm whose guarantee depends smoothly on $\gamma$; in particular, when $\gamma=1$ (the DR-submodular case) our bound recovers the $0.401$ approximation factor, while for $\gamma<1$ the guarantee degrades gracefully and, it improves upon previously reported bounds for $\gamma$-weakly DR-submodular maximization under the same constraints. Our approach combines a Frank-Wolfe-guided continuous-greedy framework with a $\gamma$-aware double-greedy step, yielding a simple yet effective procedure for handling non-monotonicity. This results in state-of-the-art guarantees for non-monotone $\gamma$-weakly DR-submodular maximization over down-closed convex bodies.