Submodular Maximization over Multiple Matroids via Generalized Exchange Properties
Submodular Maximization over Multiple Matroids via Generalized Exchange Properties
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DOI：
10.1287/moor.1100.0463
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发表时间：
2009-08
期刊：
影响因子：
--
通讯作者：
Jon Lee;M. Sviridenko;J. Vondrák
中科院分区：
文献类型：
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作者：
Jon Lee;M. Sviridenko;J. Vondrák
最大化的函数是组合优化的中心问题，概括了许多重要的NP问题，包括最大值，图形和某些约束满意度问题；任何K≥2和任何E> 0，都有一种自然的本地搜索算法，其近似保证为1/（k + e），以最大化单调的问题在1/（k + 1）的情况下，k Matroid约束的函数受到了Fisher，Nemhauser和Wolsey的影响，并于1978年获得。近似于subsodular set函数 - ii。一般的非单身酮下调函数受K Matroid约束的约束。 ） + e），我们的分析是基于两个新的Matroid的交换属性。路口。
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 e > 0, there is a natural local search algorithm that has approximation guarantee of 1/(k + e) 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 + e) and 1/(k + 1 + 1/(k-1) + e), 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.