Submodular Maximization over Multiple Matroids via Generalized Exchange Properties

• Published in: Mathematics of operations research Vol. 35; no. 4; pp. 795 - 806
• Main Authors: Lee, Jon, Sviridenko, Maxim, Vondrák, Jan
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.11.2010; Institute for Operations Research and the Management Sciences; Inst
• Subjects: Algorithms; Analysis; Approximation; approximation algorithm; Approximation algorithms; Approximation theory; Approximations; Linear programming; Mathematical functions; Mathematical sets; Mathematical theorems; matroid; Matroids; Optimal solutions; Optimization algorithms; Studies; submodular function; Vertices
• ISSN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.1100.0463

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 > 0, there is a natural local search algorithm that has approximation guarantee of 1/( k + ) 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 obtained in 1978 [Fisher, M., G. Nemhauser, L. Wolsey. 1978. An analysis of approximations for maximizing submodular set functions-II. Math. Programming Stud. 8 73-87]. 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 + ) and 1/( k + 1 + 1/( k − 1) + ), 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.