Lee, Jon, Mirrokni, Vahab S., Nagarajan, Viswanath and Sviridenko, Maxim. (2010) Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM Journal on Discrete Mathematics, Vol.23 (No.4). pp. 2053-2078. ISSN 0895-4801

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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 nonnegative 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 $(\frac{1}{k+2+\frac{1}{k}+\epsilon})$-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 $k\geq2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon})$-approximation for maximizing a monotone submodular function subject to $k\geq2$ partition matroids, which is an improvement over the previously best known guarantee of $\frac{1}{k+1}$.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions:	Faculty of Science > Computer Science
Journal or Publication Title:	SIAM Journal on Discrete Mathematics
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0895-4801
Date:	2010
Volume:	Vol.23
Number:	No.4
Page Range:	pp. 2053-2078
Identification Number:	10.1137/090750020
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/42395

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