Random Bichromatic Matchings

Abstract

Given a graph with edges colored Red and Blue, we study the problem of sampling and approximately counting the number of matchings with exactly kRed edges. We solve the problem of estimating the number of perfect matchings with exactly kRed edges for dense graphs. We study a Markov chain on the space of all matchings of a graph that favors matchings with kRed edges. We show that it is rapidly mixing using non-traditional canonical paths that can backtrack. We show that this chain can be used to sample matchings in the 2-dimensional toroidal lattice of any fixed size l with kRed edges, where the horizontal edges are Red and the vertical edges are Blue.