# Counting and Sampling *H*-Colourings

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## Abstract

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.

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