Random generation of combinatorial structures from a uniform
Theoretical Computer Science
A very simple algorithm for estimating the number of k-colorings of a low-degree graph
Random Structures & Algorithms
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
The complexity of choosing an H-colouring (nearly) uniformly at random
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Foundations of Cryptography: Basic Tools
Foundations of Cryptography: Basic Tools
Counting without sampling: new algorithms for enumeration problems using statistical physics
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Random Structures & Algorithms
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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For counting problems in #P which are "essentially self-reducible," it is known that sampling and approximate counting are equivalent. However, many problems of interest do not have such a structure and there is already some evidence that this equivalence does not hold for the whole of #P. An intriguing example is the class of H-colouring problems, which have recently been the subject of much study, and their natural generalisation to vertex- and edge-weighted versions. Particular cases of the counting-to-sampling reduction have been observed, but it has been an open question as to how far these reductions might extend to any H and a general graph G. Here we give the first completely general counting-to-sampling reduction. For every fixed H, we show that the problem of approximately determining the partition function of weighted H-colourings can be reduced to the problem of sampling these colourings from an approximately correct distribution. In particular, any rapidly mixing Markov chain for sampling H-colourings can be turned into an FPRAS for counting H-colourings.