We present combinatorial and parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to $0.193 - \varepsilon$. The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in $O( \log(n) )$ adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio $1/6 - \varepsilon$, adaptivity $O( \log (n) )$, and query complexity $O( n \log (k) )$, while the second has approximation ratio $0.193 - \varepsilon$, adaptivity $O( \log^2 (n) )$, and query complexity $O(n \log (k))$. Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.

翻译：我们提出了用于最大化子模函数（不一定是单调的）的组合可并行算法，该函数受大小约束。我们将具有最优自适应性和近最优查询复杂度的算法所达到的最佳近似因子改进至$0.193 - \varepsilon$。本工作的会议版本错误地采用了一个不适用于非单调子模函数的子程序。在此版本中，我们提出一个经过修正与改进的子程序ThreshSeq，用于添加具有高平均边际增益的集合，该子程序以高概率在$O( \log(n) )$自适应轮次内返回解。此外，我们提供两种近似算法。第一种算法的近似比为$1/6 - \varepsilon$，自适应复杂度为$O( \log (n) )$，查询复杂度为$O( n \log (k) )$；第二种算法的近似比为$0.193 - \varepsilon$，自适应复杂度为$O( \log^2 (n) )$，查询复杂度为$O(n \log (k))$。实验验证表明，与最先进的近似算法（包括使用多线性延拓的连续算法）相比，我们的算法在保持较低自适应轮次和总查询次数的同时，能获得具有高目标值的解。