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

翻译：随着现代应用中数据的快速增长，用于最大化非单调子模函数的并行算法已受到广泛关注。在并行计算设置中，当前最优的1/e近似比由一种自适应复杂度为$ O\left(\log(n)\right)$的连续算法实现（Ene & Nguyen, 2020）。本文聚焦于基数约束问题，提出了首个达到该边界的组合算法——一种随机化并行方法，实现了$1/e-\varepsilon$的近似比。该结果弥合了该问题连续方法与组合方法之间的鸿沟。作为副产品，我们还开发了一种更简单的、以高概率（$\ge 1-1/n$）达到$(1/4-\varepsilon)$近似度的算法。两种算法均实现了$ O\left(\log(n)\log(k)\right)$的自适应复杂度与$O\left(n\log(n)\log(k)\right)$的查询复杂度。实验结果表明，我们的算法能获得具有竞争力的目标函数值，其中$(1/4-\varepsilon)$近似算法在查询效率方面表现尤为突出。