Breaking Barriers: Combinatorial Algorithms for Non-monotone Submodular Maximization with Sublinear Adaptivity and $1/e$ Approximation

TL;DR

This paper introduces the first combinatorial parallel algorithms for non-monotone submodular maximization with size constraints, achieving near state-of-the-art approximation ratios and low adaptivity, bridging the gap with continuous methods.

Contribution

It presents a novel combinatorial parallel algorithm matching the $1/e$ approximation ratio with $O( ext{log}(n))$ adaptivity, and a simpler $(1/4- ext{epsilon})$-approximation algorithm, advancing the field.

Findings

Achieves $1/e- ext{epsilon}$ approximation with combinatorial approach.

Both algorithms have $O( ext{log}(n) ext{log}(k))$ adaptivity.

Empirical results show competitive objective values and efficiency.

Abstract

With the rapid growth of data in modern applications, parallel algorithms for maximizing non-monotone submodular functions have gained significant attention. In the parallel computation setting, the state-of-the-art approximation ratio of $1/e$ is achieved by a continuous algorithm (Ene & Nguyen, 2020) with adaptivity $ O\left(\log(n)\right)$. In this work, we focus on size constraints and present the first combinatorial algorithm matching this bound -- a randomized parallel approach achieving $1/e-\varepsilon$ approximation ratio. This result bridges the gap between continuous and combinatorial approaches for this problem. As a byproduct, we also develop a simpler $(1/4-\varepsilon)$-approximation algorithm with high probability ($\ge 1-1/n$). Both algorithms achieve $ O\left(\log(n)\log(k)\right)$ adaptivity and $O\left(n\log(n)\log(k)\right)$ query complexity. Empirical results show our algorithms achieve competitive objective values, with the $(1/4-\varepsilon)$-approximation algorithm particularly efficient in queries.