Matching Algorithms Are Fast in Sparse Random Graphs

  • Holger Bast
  • Kurt Mehlhorn
  • Guido Schäfer
  • Hisao Tamaki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2996)


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 \(O(m\sqrt{n})\), 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), 1994]. Our results hold, if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.


Bipartite Graph Random Graph General Graph Maximum Match Expander Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Holger Bast
    • 1
  • Kurt Mehlhorn
    • 1
  • Guido Schäfer
    • 1
  • Hisao Tamaki
    • 2
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Meiji UniversityKawasakiJapan

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