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.

翻译：子模最大化是组合优化中的核心课题之一，在现实世界中有着众多应用。过去几十年中，针对这一问题已提出了一系列算法。然而，当前最先进的算法大多为随机性算法。在子模最大化问题中，确定性算法与随机算法在近似比方面仍存在不可忽视的差距。本文针对非单调子模最大化问题，提出了具有改进近似比的确定性算法。具体而言，对于拟阵约束，我们给出了一种确定性$0.283-o(1)$近似算法，而此前最优确定性算法仅能达到$1/4$的近似比；对于背包约束，我们给出了一种确定性$1/4$近似算法，而此前最优确定性算法仅能达到$1/6$的近似比；对于大宽度线性装箱约束，我们给出了一种确定性$1/6-\epsilon$近似算法。据我们所知，目前针对这些约束尚不存在确定性近似算法。