Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints
This work addresses a gap in combinatorial optimization for researchers and practitioners needing deterministic algorithms, offering incremental improvements over prior deterministic state-of-the-art ratios.
The paper tackles the problem of deterministic approximation algorithms for maximizing non-monotone submodular functions under matroid and knapsack constraints, achieving approximation ratios of (0.385 - ε) for matroid and (0.367 - ε) for knapsack constraints with polynomial query complexity.
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 $(0.385 - ε)$, whereas for knapsack constraints, the proposed algorithm attains an approximation ratio of $(0.367 -ε)$. 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 $(0.367 - ε)$ for matroid constraints and $0.25$ for knapsack constraints.