Article

Algorithmica

, Volume 50, Issue 4, pp 418-445

Random Bichromatic Matchings

  • Nayantara BhatnagarAffiliated withCollege of Computing, Georgia Institute of Technology Email author 
  • , Dana RandallAffiliated withCollege of Computing, Georgia Institute of Technology
  • , Vijay V. VaziraniAffiliated withCollege of Computing, Georgia Institute of Technology
  • , Eric VigodaAffiliated withCollege of Computing, Georgia Institute of Technology

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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 k Red edges. We solve the problem of estimating the number of perfect matchings with exactly k Red edges for dense graphs. We study a Markov chain on the space of all matchings of a graph that favors matchings with k Red 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 with k Red edges, where the horizontal edges are Red and the vertical edges are Blue.

Keywords

Markov chains Matchings Sampling Approximate counting