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Rapidly mixing Markov chains for dismantleable constraint graphs

By Martin Dyer, Mark Jerrum and Eric Vigoda


If G = (VG, EG) is an input graph, and H = (VH, EH) a fixed constraint graph, we study the set Ω of homomorphisms (or colorings) from VG to VH, i.e. functions which preserve adjacency. Brightwell and Winkler introduced the notion of dismantleable constraint graph to characterize those H whose set Ω is connected under single vertex recolorings for every G. Given fugacities λ(c)> 0 (c ∈ VH) our focus is on sampling an ω ∈ Ω according to the Gibbs distribution, i.e., with probability proportional to ∏ λ(ω(v)). We prove, for each v∈VG dismantleable H, that there exist positive constant fugacities on VH such that the Glauber dynamics, a Markov chain which recolors a single vertex at each step, has mixing time O(n2) for all bounded degree graphs G

Year: 2001
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