Counting and Sampling H-Colourings
- 711 Downloads
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.
Unable to display preview. Download preview PDF.
- 1.G.R. Brightwell and L.A. Goldberg, personal communication.Google Scholar
- 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
- 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.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.L.A. Goldberg, M. Jerrum and M. Paterson, The computational complexity of two-state spin systems, Pre-print (2001).Google Scholar
- 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.O. Goldreich, The Foundations of Cryptography-Volume 1, (Cambridge University Press, 2001)Google Scholar
- 12.M. Jerrum, Sampling and Counting. Chapter 3 of Counting, Sampling and Integrating: Algorithms and Complexity, Birkhäuser, Basel. (In preparation.)Google Scholar
- 14.C.H. Papadimitriou, Computational Complexity, (Addison-Wesley, 1994)Google Scholar