Non-monotone submodular maximization 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 ( 1/k+2+1/k +ε)-approximation for the submodular maximization problem under k matroid constraints, and a (1/5 - ε)-approximation algorithm for this problem subject to k knapsack constraints (ε > 0 is any constant). We improve the approximation guarantee of our algorithm to 1/k+1+1/k-1+ε for k ≥ 2 partition matroid constraints. This idea also gives a ( 1/k+ε)- approximation for maximizing a monotone submodular function subject to k ≥ 2 partition matroids, which improves over the previously best known guarantee of 1/k+1. Copyright 2009 ACM.