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

Yixin Chen, Wenjing Chen, Alan Kuhnle

TL;DR: First combinatorial algorithm for the non-monotone submodular maximization problem achieving a $1/e$ approximation ratio with sublinear adaptivity

Abstract:

With the rapid growth of data in modern applications, parallel combinatorial 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 $\mathcal O (log(n))$. In this work, we focus on size constraints and present the first combinatorial algorithm matching this bound – a randomized parallel approach achieving $1/e − \epsilon$ approximation ratio. This result bridges
the gap between continuous and combinatorial approaches for this problem. As a byproduct, we also develop a simpler $(1/4 − \epsilon)$-approximation algorithm with high probability $(\ge 1 − 1/n)$. Both algorithms achieve $\mathcal O (log(n) log(k))$ adaptivity and $\mathcal O (n log(n) log(k)) query complexity. Empirical results show our algorithms achieve competitive objective values, with the $(1/4 − \epsilon)$-approximation algorithm particularly efficient in queries.

Lay Summary: First practical, parallelizable algorithms for size-constrained maximization of submodular functions that achieve theoretical performance guarantees of $1/e$.

Link To Code: https://gitlab.com/luciacyx/size-constraints-parallel-algorithms.git

Primary Area: Optimization->Discrete and Combinatorial Optimization

Keywords: submodular optimization, combinatorial algorithms, parallel algorithms

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Submission Number: 7346