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Counting and Sampling H-Colourings

  • Martin Dyer
  • Leslie A. Goldberg
  • Mark Jerrum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Martin Dyer
    • 1
  • Leslie A. Goldberg
    • 2
  • Mark Jerrum
    • 3
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryUK
  3. 3.Division of InformaticsUniversity of EdinburghEdinburghUK

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