Gibbs measures and dismantlable graphs
Journal of Combinatorial Theory Series B
The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and algorithms
On Markov chains for randomly H-coloring a graph
Journal of Algorithms
On the relative complexity of approximate counting problems
APPROX '00 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization
Counting and Sampling H-Colourings
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Counting and Sampling H-Colourings
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
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If G = (VG, EG) is an input graph, and H = (VH, EH) a fixed constraint graph, we study the set 驴 of homomorphisms (or colorings) from VG to VH, i.e., functions that preserve adjacency. Brightwell and Winkler introduced the notion of dismantleable constraint graph to characterize those H whose associated set 驴 of homomorphisms is, for every G, connected under single vertex recolorings. Given fugacities 驴(c) 0 (c 驴 VH) our focus is on sampling a coloring &omega 驴 驴 according to the Gibbs distribution, i.e., with probability proportional to 驴v驴VG 驴(驴(v)). The Glauber dynamics is a Markov chain on 驴 which recolors a single vertex at each step, and leaves invariant the Gibbs distribution. We prove that, for each dismantleable H and degree bound 驴, there exist positive constant fugacities on VH such that the Glauber dynamics has mixing time O(n2), for all graphs G whose vertex degrees are bounded by 驴.