We study parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a cardinality constraint $k$. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and query complexity, up to logarithmic factors in the size $n$ of the ground set, from $0.039 - \epsilon$ to $0.193 - \epsilon$. We provide two algorithms; the first has approximation ratio $1/6 - \epsilon$, adaptivity $O( \log n )$, and query complexity $O( n \log k )$, while the second has approximation ratio $0.193 - \epsilon$, adaptivity $O( \log^2 n )$, and query complexity $O(n \log k)$. Heuristic versions of 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:我们研究的是使亚模量函数最大化的平行算法,但不一定是单调法,对于基质限制,我们研究的是美元。我们改进了最优化适应性和查询复杂性的算法所达到的最佳近似系数,从0.039美元到0.193美元,最高可达地面数的对数系数,从0.39美元到0.193美元。我们提供了两种算法;我们提供了两种算法;第一种算法:近似比率1/6美元 -\ epsilon$,适应性$O(n\log n),和查询复杂度$O(n\log k)$,而第二种算法则有0.193-\ explon$,适配值$O(\log2 n)和查询复杂性$(n\log k)美元。我们的算法的超标准版本经过经验验证,可以使用少量的适应性圆和总查询,同时获得与最新近似近似算算法相比具有高客观价值的解决办法,包括使用多线扩展的连续算法。