Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs
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If G = (V G ,E G ) is an input graph, and H = (V h,Eh) a fixed constraint graph, we study the set Ω of homomorphisms (or colorings) from V g to V h, i.e., functions that preserve adjacency. Brightwell and Winkler introduced the notion of dismantleable constraint graph to characterize those H whose associated set Ω of homomorphisms is, for every G, connected under single vertex recolorings. Given fugacities λ(c) > 0 (c ∈ V h) our focus is on sampling a coloring ω ∈ Ω according to the Gibbs distribution, i.e., with probability proportional to Πυ∈V Gλ(ω(υ)). The Glauber dynamics is a Markov chain on Ω which recolors a single vertex at each step, and leaves invariant the Gibbs distribution. We prove that, for each dismantleable H and degree bound Δ, there exist positive constant fugacities on V h such that the Glauber dynamics has mixing time O(n 2), for all graphs G whose vertex degrees are bounded by Δ.
KeywordsMarkov Chain Stationary Distribution Variant Dynamic Markov Chain Monte Carlo Method Input Graph
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