# Maximizing nonmonotone submodular functions under matroid or knapsack constraints

## Abstract

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 nonneg-ative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone submodular functions. In particular, for any constant k, we present a (1/k+2+1/k+η-)- approximation for the submodular maximization problem under k matroid constraints, and a (1/5 - η)-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+η for k ≥ 2 partition matroid constraints. This idea also gives a (1/k+η)-approximation for maximizing a monotone submodular function subject to k ≥ 2 partition matroids, which is an improvement over the previously best known guarantee of 1/k+1 . © 2009 Society for Industrial and Applied Mathematics.