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

Abstract

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 H-colourings. 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 Randomised Approximation Scheme (FPRAS) for BIS --- the problem of counting independent sets in bipartite graphs. Dyer, Goldberg, Greenhill and Jerrum have shown that BIS is complete in a certain logically-defined complexity class. Thus, a PAUS for sampling H-colourings would give an FPRAS for the entire complexity class. In order to achieve our result we introduce the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserving reduction.