Maximizing Non-monotone Submodular Functions under Matroid and Knapsack Constraints

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a -approximation for the submodular maximization problem under k matroid constraints, and a -approximation algorithm for this problem subject to k knapsack constraints ( is any constant). We improve the approximation guarantee of our algorithm to for partition matroid constraints. This idea also gives -approximation for maximizing a monotone submodular function subject to partition matroids, which improves over the previously best known guarantee of