NDLI: Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Nagarajan, Viswanath Sviridenko, Maxim Mirrokni, Vahab S. Lee, Jon
Copyright Year	2010
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 $\frac{1}{k+1}$.
Starting Page	2053
Ending Page	2078
Page Count	26
File Format	PDF
ISSN	08954801
DOI	10.1137/090750020
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	23
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-01-15
Access Restriction	Subscribed
Subject Keyword	submodular maximization matroid and knapsack constraints Approximation algorithms Combinatorial optimization approximationalgorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics

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