Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms - Gupta, Anupam; Roth, Aaron; Schoenebeck, Grant; Talwar, Kunal (Internet and Network Economics, 2010)
Constrained submodular maximization problems have long been studied, most recently in the context of auctions and computational advertising, with near-optimal results known under a variety of constraints when the submodular function is monotone. In this paper, we give constant approximation algorithms for the non-monotone case that work for p-independence systems (which generalize constraints given by the intersection of p matroids that had been studied previously), where the running time is $\text{poly}(n,p)$ . Our algorithms and analyses are simple, and essentially reduce non-monotone maximization to multiple runs of the greedy algorithm previously used in the monotone case. We extend these ideas to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting (where elements arrive one at a time in random order and the algorithm must make irrevocable decisions) subject to uniform matroid or a partition matroid constraint. Finally, we give an O(logk) approximation in the secretary setting subject to a general matroid constraint of rank k.