Counting and Sampling H-Colourings

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.

This work was partially supported by the EPSRC grant “Sharper Analysis of Randomised Algorithms: a Computational Approach”, the EPSRC grant GR/R44560/01 “Analysing Markov-chain based random sampling algorithms” and the IST Programme of the EU under contract numbers IST-1999-14186 (ALCOM-FT) and IST-1999-14036 (RAND-APX).