Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs

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.

This work was supported by EPSRC Research grant “Sharper Analysis of Randomised Algorithms: A Computational Approach” and in part by the ESPRIT Project RAND-APX. The last author was partially supported by a Koshland Scholar award from the Weizmann Institute of Science.