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)


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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  1. 1.
    G.R. Brightwell and L.A. Goldberg, personal communication.Google Scholar
  2. 2.
    G.R. Brightwell and P. Winkler, Gibbs measures and dismantlable graphs, J. Com-bin. Theory Ser. B 78(1) 141–166 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    C. Cooper, M. Dyer and A. Frieze, On Markov chains for randomly H-colouring a graph, Journal of Algorithms, 39(1) (2001) 117–134.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Dyer and C. Greenhill, Random walks on combinatorial objects. In J.D. Lamb and D.A. Preece, editors, Surveys in Combinatorics, volume 267 of London Mathematical Society Lecture Note Series, pages 101–136. Cambridge University Press, 1999.Google Scholar
  5. 5.
    M. Dyer and C. Greenhill, The complexity of counting graph homomorphisms. Random Structures and Algorithms, 17 (2000) 260–289.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Dyer, M. Jerrum and E. Vigoda, Rapidly mixing Markov chains for disman-tleable constraint graphs. In J. Nesetril and P. Winkler, editors, Proceedings of a DIM ACS/DIMATI A Workshop on Graphs, Morphisms and Statistical Physics, March 2001, to appear.Google Scholar
  7. 7.
    L.A. Goldberg, Computation in permutation groups: counting and randomly sampling orbits. In J.W.P. Hirschfeld, editor, Surveys in Combinatorics, volume 288 of London Mathematical Society Lecture Note Series, pages 109–143. Cambridge University press, 2001.Google Scholar
  8. 8.
    L.A. Goldberg, M. Jerrum and M. Paterson, The computational complexity of two-state spin systems, Pre-print (2001).Google Scholar
  9. 9.
    L.A. Goldberg, S. Kelk and M. Paterson, The complexity of choosing an H- colouring (nearly) uniformly at random, To appear in STOC 2002.Google Scholar
  10. 10.
    O. Goldreich, The Foundations of Cryptography-Volume 1, (Cambridge University Press, 2001)Google Scholar
  11. 11.
    M. Jerrum, A very simple algorithm for estimating the number of k-colorings of a low-degree graph, Random Structures and Algorithms, 7 (1995) 157–165.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Jerrum, Sampling and Counting. Chapter 3 of Counting, Sampling and Integrating: Algorithms and Complexity, Birkhäuser, Basel. (In preparation.)Google Scholar
  13. 13.
    M.R. Jerrum, L.G. Valiant, and V.V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science, 43 (1986) 169–188.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    C.H. Papadimitriou, Computational Complexity, (Addison-Wesley, 1994)Google Scholar

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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