The problems of maximizing constrained monotone submodular functions have many practical applications, most recently in the context of combinatorial optimization, operations research, economics and especially machine learning, with constant approximation algorithms known under a variety of constraints. Unfortunately, non-monotone submodular functions maximization is less well studied; the first approximation algorithm for the non-monotone case was studied by Feige et al. (Proceedings of the 48th IEEE symposium on foundations of computer science (FOCS’07), 2007) about unconstrained non-monotone submodular maximization in 2007. In this paper, we extend the work of Lee et al. (Proceedings of the 41st ACM-SIAM symposium on theory of computing (STOC’09), pp 323–332, 2009) for maximizing a non-monotone submodular function under k-matroid constraint to k-system constraint. We first propose a Modified-Greedy algorithm that works no worse than that of Gupta et al. (Proceedings of the 6th international workshop on internet and network economics (WINE’10), vol 6484, pp 246–257, 2010). Based on this, then we provide the NMSFMk algorithm for maximizing a non-monotone submodular function subject to k-system constraint (which generalizes the k-matroid constraint), using Modified-Greedy algorithm combined with USFM algorithm (USFM algorithm is the random linear time 1/2-approximation algorithm proposed by Buchbinder et al. (Proceedings of the 53rd IEEE symposium on foundations of computer science (FOCS’12), pp 649–658, 2012) for unconstrained non-monotone submodular function maximization problem.) iteratively. Finally, we show that NMSFMk algorithm achieves a \(\frac{1}{2k+3+1/k}\)-approximation ratio with running time of O(nmk) (where m is the size of largest set returned by the NMSFMk algorithm), which beats the existing algorithms in many aspects.

最大化约束单调子模函数的问题有许多实际应用，最近在组合优化，运筹学，经济学尤其是机器学习的背景下，采用了在各种约束下已知的恒定逼近算法。不幸的是，对非单调子模函数最大化的研究较少。Feige等人研究了非单调情况的第一种近似算法。（第48届IEEE计算机科学基础学术研讨会论文集（FOCS'07），2007年），涉及2007年无约束非单调子模极大值。本文扩展了Lee等人的工作。（第41届ACM-SIAM计算理论研讨会论文集（STOC'09），第323-332页，2009年），该文旨在在以下条件下最大化非单调子模函数k -matroid约束到k-系统约束。我们首先提出一种Modified-Greedy算法，其工作原理不比Gupta等人的算法差。（第六届国际互联网和网络经济学研讨会论文集（WINE'10），第6484卷，第246–257页，2010年）。基于此，我们提供了NMSFMk算法，用于最大化受k-系统约束的非单调子模函数（将k-matroid约束），结合使用Modified-Greedy算法和USFM算法（USFM算法是Buchbinder等人提出的随机线性时间1/2逼近算法）（第53届IEEE计算机科学基础学术研讨会论文集（FOCS'12 ），第649–658页，2012），用于无约束非单调子模函数最大化问题。最后，我们证明NMSFMk算法在运行时间为O（nmk）的情况下达到了\（\ frac {1} {2k + 3 + 1 / k} \） -逼近比（其中m是由NMSFMk算法），它在许多方面都优于现有算法。