Maximum matching in graphs with an excluded minor

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

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