# Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs

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## Abstract

If *G = (V* _{ G },*E* _{ G }) is an input graph, and *H = (V* _{h},E_{h}) 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 Π_{υ∈V} _{G}λ(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.

## Keywords

Markov Chain Stationary Distribution Variant Dynamic Markov Chain Monte Carlo Method Input Graph## Preview

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