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Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs

  • Martin Dyer
  • Mark Jerrum
  • Eric Vigoda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)

Abstract

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.

Keywords

Markov Chain Stationary Distribution Variant Dynamic Markov Chain Monte Carlo Method Input Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Martin Dyer
    • 1
  • Mark Jerrum
    • 2
  • Eric Vigoda
    • 3
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.Laboratory for Foundations of Computer ScienceUniversity of EdinburghEdinburghUK
  3. 3.Department of Computer ScienceUniversity of ChicagoChicagoUSA

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