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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 $\ln (n)$ [\emph{Average Case
Analysis of Algorithms for Matchings and Related Problems}, Journal of the ACM,
\textbf{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.
