ALCOMFT-TR-03-111

Holger Bast, Kurt Mehlhorn, Guido Schäfer and Hisao Tamaki
Matching Algorithms are Fast in Sparse Random Graphs
MPI. Work package 4. November 2003.

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\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 result holds, if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.

Postscript file: ALCOMFT-TR-03-111.ps.gz (95 kb).