Robust matchings and matroid intersections

Authors:
Ryo Fujita;Yusuke Kobayashi;Kazuhisa Makino
Affiliations:
Cannon Inc., Tokyo, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, University of Tokyo, Tokyo
Venue:
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
Year:
2010

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Abstract

In a weighted undirected graph, a matching is said to be α-robust if for all p, the total weight of its heaviest p edges is at least α times the maximum weight of a p-matching in the graph. Here a p-matching is a matching with at most p edges. In 2002, Hassin and Rubinstein [4] showed that every graph has a 1/√2-robust matching and it can be found by k-th power algorithm in polynomial time. In this paper, we show that it can be extended to the matroid intersection problem, i.e., there always exists a 1/√2-robust matroid intersection, which is polynomially computable. We also study the time complexity of the robust matching problem. We show that a 1-robust matching can be computed in polynomial time (if exists), and for any fixed number α with 1/√2k-th power algorithm for robust matchings, i.e., for any ε 0, there exists a weighted graph such that no k-th power algorithm outputs a (1/√2+ ε)-approximation for computing the most robust matching.