Theory of Computing Systems

, Volume 39, Issue 1, pp 3–14

Matching Algorithms Are Fast in Sparse Random Graphs

Article

Abstract

We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft-Karp algorithm for bipartite graphs and the Micali-Vazirani algorithm for general graphs, which have a worst case running time of O(m√n), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln (n) [Average Case Analysis of Algorithms for Matchings and Related Problems, Journal of the ACM, 41(6):1329-1356, 1994]. Our results hold if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Max-Planck-Institut fur Informatik, Stuhlsatzenhausweg 85, 66123 SaarbruckenGermany
  2. 2.Meiji University, School of Science and Technology, 1-1-1 HigashiMita, Tama, Kawasaki 214-8571Japan

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