TY - GEN

T1 - Maximum matching in graphs with an excluded minor

AU - Yuster, Raphael

AU - Zwick, Uri

N1 - Publisher Copyright:
Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics.

PY - 2007

Y1 - 2007

N2 - 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(n3ω/(ω+3)) < O(n1.326) 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(n1.5) time bound obtained by applying the O(mn1/2)-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(nω/2) < O(n1.19) 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(n1+ω/2) < O(n2.19) 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.

AB - 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(n3ω/(ω+3)) < O(n1.326) 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(n1.5) time bound obtained by applying the O(mn1/2)-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(nω/2) < O(n1.19) 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(n1+ω/2) < O(n2.19) 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.

UR - http://www.scopus.com/inward/record.url?scp=84969165496&partnerID=8YFLogxK

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AN - SCOPUS:84969165496

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 108

EP - 117

BT - Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007

PB - Association for Computing Machinery

T2 - 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007

Y2 - 7 January 2007 through 9 January 2007

ER -