Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

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Article ID: iaor20108768
Volume: 35
Issue: 4
Start Page Number: 795
End Page Number: 806
Publication Date: Nov 2010
Journal: Mathematics of Operations Research
Authors: , ,
Keywords: graphs
Abstract:

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including max cut in digraphs, graphs, and hypergraphs; certain constraint satisfaction problems; maximum entropy sampling; and maximum facility location problems. Our main result is that for any k ≥ 2 and any ϵ > 0, there is a natural local search algorithm that has approximation guarantee of 1/(k + ϵ) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves upon the 1/(k + 1)-approximation of Fisher, Nemhauser, and Wolsey (1978). Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general nonmonotone submodular function subject to k matroid constraints. We show that, in these cases, the approximation guarantees of our algorithms are 1/(k-1 + ϵ) and 1/(k + 1 + 1/(k-1) + ϵ), respectively.Our analyses are based on two new exchange properties for matroids. One is a generalization of the classical Rota exchange property for matroid bases, and another is an exchange property for two matroids based on the structure of matroid intersection.

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