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 + " -approximation for the submodular maximization problem under k matroid constraints, and a `1 5 ? -approximation algorithm for this problem subject to k knapsack constraints ( > 0 is any constant). We improve the approximation guarantee of our algorithm to 1 k+1+ 1 k-1 + for k 2 partition matroid constr...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, Maxim Sviridenko
• Keywords: Algorithms, Knapsack Constraints, Partition Matroid Constraints, STOC 2009, Submodular Maximization Problem