Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

Sulmodular function maximization is a central problem in combinatorial optimization. generalizing litany 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 ally non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are fer non-monotone submodular functions. In particular, for any constant k. we present, a (1/k+2+1/k+epsilon)-approximation for the submodular maximization problem under k matroid constraints, and a (1/5 - epsilon)-approximation algorithm for this problem subject to k knapsack constraints (epsilon > 0 is any constant). We improve the approximation guarantee of out algorithm to 1/k+1+1/k-1+epsilon) for k >= 2 partition matroid constrants. This idea also gives a (1/k+epsilon)-approximation for maximizing a monotone submoldular function subject to k >= 2 partition matroids, which improves over the previously best known guarantee of 1/k+1.