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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Bast, H., Mehlhorn, K., Schafer, G. et al. Matching Algorithms Are Fast in Sparse Random Graphs. Theory Comput Syst 39, 3–14 (2006). https://doi.org/10.1007/s00224-005-1254-y
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DOI: https://doi.org/10.1007/s00224-005-1254-y