Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

TL;DR

This paper introduces new deterministic algorithms for non-monotone submodular maximization under matroid and knapsack constraints, achieving improved approximation ratios over previous deterministic methods.

Contribution

The authors develop novel deterministic algorithms based on an extended multilinear extension framework for non-monotone submodular maximization under two key constraints.

Findings

Achieves a $(0.385 - \epsilon)$ approximation for matroid constraints.

Achieves a $(0.367 - \\epsilon)$ approximation for knapsack constraints.

Runs in polynomial query complexity, improving previous deterministic bounds.

Abstract

Submodular maximization constitutes a prominent research topic in combinatorial optimization and theoretical computer science, with extensive applications across diverse domains. While substantial advancements have been achieved in approximation algorithms for submodular maximization, the majority of algorithms yielding high approximation guarantees are randomized. In this work, we investigate deterministic approximation algorithms for maximizing non-monotone submodular functions subject to matroid and knapsack constraints. For the two distinct constraint settings, we propose novel deterministic algorithms grounded in an extended multilinear extension framework. Under matroid constraints, our algorithm achieves an approximation ratio of $(0.385 - \epsilon)$, whereas for knapsack constraints, the proposed algorithm attains an approximation ratio of $(0.367 -\epsilon)$. Both algorithms run in $\mathrm{poly}(n)$ query complexity, where $n$ is the size of the ground set, and improve upon the state-of-the-art deterministic approximation ratios of $(0.367 - \epsilon)$ for matroid constraints and $0.25$ for knapsack constraints.