L.A. Goldberg, S. Kelk and M. Paterson, The complexity of choosing an H-colouring (nearly) uniformly at random (August 23, 2001).
Cooper, Dyer and Frieze studied the problem of sampling H-colourings (nearly) uniformly at random. They 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.