## Abstract

We present a new randomized algorithm for finding a maximum matching in H-minor free graphs. For every fixed H, our algorithm runs in O(n^{3ω/(ω+3)}) < O(n^{1.326}) time, where n is the number of vertices of the input graph and ω < 2.376 is the exponent of matrix multiplication. This improves upon the previous O(n^{1.5}) time bound obtained by applying the O(mn^{1/2})-time algorithm of Micali and Vazirani on this important class of graphs. For graphs with bounded genus, which are special cases of H-minor free graphs, we present a randomized algorithm for finding a maximum matching in O(n^{ω/2}) < O(n^{1.19}) time. This extends a previous randomized algorithm of Mucha and Sankowski, having the same running time, that finds a maximum matching in a planar graphs. We also present a deterministic algorithm with a running time of O(n^{1+ω/2}) < O(n^{2.19}) for counting the number of perfect matchings in graphs with bounded genus. This algorithm combines the techniques used by the algorithms above with the counting technique of Kasteleyn. Using this algorithm we can also count, within the same running time, the number of T-joins in planar graphs. As special cases, we get algorithms for counting Eulerian subgraphs (T = φ) and odd subgraphs (T = V) of planar graphs.

Original language | English |
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Title of host publication | Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 |

Publisher | Association for Computing Machinery |

Pages | 108-117 |

Number of pages | 10 |

ISBN (Electronic) | 9780898716245 |

State | Published - 2007 |

Event | 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 - New Orleans, United States Duration: 7 Jan 2007 → 9 Jan 2007 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 07-09-January-2007 |

### Conference

Conference | 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 |
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Country/Territory | United States |

City | New Orleans |

Period | 7/01/07 → 9/01/07 |

### Bibliographical note

Funding Information:Research supported by a GIF grant.

Publisher Copyright:

Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics.

## ASJC Scopus subject areas

- Software
- Mathematics (all)