Improved Deterministic Algorithms for Non-monotone Submodular Maximization

TL;DR

This paper introduces improved deterministic algorithms for non-monotone submodular maximization under various constraints, narrowing the gap with randomized methods and achieving better approximation ratios.

Contribution

It presents the first deterministic algorithms with enhanced approximation ratios for non-monotone submodular maximization under matroid, knapsack, and linear packing constraints.

Findings

Deterministic 0.283-o(1) approximation for matroid constraints.

Deterministic 1/4 approximation for knapsack constraints.

Deterministic 1/6-epsilon approximation for linear packing constraints.

Abstract

Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization. In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic $0.283-o(1)$ approximation algorithm, while the previous best deterministic algorithm only achieves a $1/4$ approximation ratio. For the knapsack constraint, we provide a deterministic $1/4$ approximation algorithm, while the previous best deterministic algorithm only achieves a $1/6$ approximation ratio. For the linear packing constraints with large widths, we provide a deterministic $1/6-\epsilon$ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints.