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+ -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 .
In the setting of constrained non-monotone submodular maximization, Lee et al. REF give a 1/( + 2 + 1/ + ) approximation subject to matroid constraints and a 1/(5 − ) approximation for knapsack constraints.
Non-monotone submodular maximization under matroid and knapsack constraints
STOC '09
Mathematics Computer Science