The complexity of choosing an H-colouring (nearly) uniformly at random

Cooper, Dyer and Frieze studied the problem of sampling H-colourings (nearly) uniformly at random. Special cases of this problem include sampling colourings and independent sets and sampling from statistical physics models such as the Widom-Rowlinson model, the Beach model, the Potts model and the hard-core lattice gas model. Cooper et al. considered the family of "cautious" ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected "nontrivial" graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any Polynomial Almost Uniform Sampler (PAUS) for Hcolourings. We show that if there were a PAUS for the H-colouring problem, there would also be a PAUS for sampling independent sets in bipartite graphs and, by the self-reducibility of the latter problem, there would be a Fully-Polynomial Ra...

• Type: Conference
• Year: 2002
• Where: STOC
• Authors: Leslie Ann Goldberg, Steven Kelk, Mike Paterson
• Keywords: Algorithms, Ergodic Markov Chains, Hard-core Lattice Gas, Statistical Physics Models, STOC 2002