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#### Matching Algorithms are Fast in Sparse Random Graphs

##### MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44076

Bast,  Holger
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45021

Mehlhorn,  Kurt
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45363

Schäfer,  Guido
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45588

Tamaki,  Hisao
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Zitation

Bast, H., Mehlhorn, K., Schäfer, G., & Tamaki, H. (2006). Matching Algorithms are Fast in Sparse Random Graphs. Theory of Computing Systems, 39, 3-14.

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.