Random generation of combinatorial structures from a uniform
Theoretical Computer Science
On the complexity of H-coloring
Journal of Combinatorial Theory Series B
A very simple algorithm for estimating the number of k-colorings of a low-degree graph
Random Structures & Algorithms
Graph homomorphisms and phase transitions
Journal of Combinatorial Theory Series B
On Markov chains for independent sets
Journal of Algorithms
The complexity of H-colouring of bounded degree graphs
Discrete Mathematics
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 Counting Independent Sets in Sparse Graphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Counting and Sampling H-Colourings
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Counting and sampling H-colourings
Information and Computation
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Classification of a Class of Counting Problems Using Holographic Reductions
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
A Complexity Dichotomy for Partition Functions with Mixed Signs
SIAM Journal on Computing
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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.