Non-monotone submodular maximization under matroid and 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. For the problem of maximizing a non-monotone submodular function, Feige, Mirrokni, and Vondrák recently developed a $\frac{2}{5}$-approximation algorithm \cite{FMV07}, however, their algorithms do not handle side constraints.} 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 {\em non-monotone} submodular functions. In particular, for any constant $k$, we present a $(\frac{1}{k+2+\frac{1}{k}+\epsilon })$-approximation for the submodular maximization problem under $k$ matroid constraints, and a $(\frac{1}{5}-\epsilon )$-approximation algorithm for this problem subject to $k$ knapsack constraints ($\epsilon >0$ is any constant). We improve the approximation guarantee of our algorithm to $\frac{1}{k+1+\frac{1}{k-1}+\epsilon }$ for $k\ge 2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon })$-approximation for maximizing a {\em monotone} submodular function subject to $k\ge 2$ partition matroids, which improves over the previously best known guarantee of $\frac{1}{k+1}$.