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Sampling Grid Colorings with Fewer Colors

  • Dimitris Achlioptas
  • Mike Molloy
  • Cristopher Moore
  • Frank Van Bussel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

We provide an optimally mixing Markov chain for 6-colorings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colorings has strong spatial mixing. Four and five are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colorings of the square grid.

Keywords

Markov Chain Gibbs Measure Graph Coloring Free Boundary Condition Glauber Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aldous, D.: Random walks on finite groups and rapidly mixing Markov chains. In: Dold, A., Eckmann, B. (eds.) Séminaire de Probabilités XVII 1981/82. Springer Lecture Notes in Mathematics, vol. 986, pp. 243–297 (1986)Google Scholar
  2. 2.
    Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proc. 28th Ann. Symp. on Found. of Comp. Sci., pp. 223–231 (1997)Google Scholar
  3. 3.
    Bubley, R., Dyer, M., Greenhill, C.: Beating the 2Δ bound for approximately counting colourings: A computer-assisted proof of rapid mixing. In: Proc. 9th Ann. ACM-SIAM Symposium on Discrete Algorithms, pp. 355–363 (1998)Google Scholar
  4. 4.
    Cesi, F.: Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probability Theory and Related Fields 120, 569–584 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible Markov chains. Annals of Applied Probability 6, 696–730 (1996)MathSciNetGoogle Scholar
  6. 6.
    Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Annals of Applied Probability 1, 36–61 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dyer, M., Greenhill, C.: A more rapidly mixing Markov chain for graph colorings. Random Structures and Algorithms 13, 285–317 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dyer, M., Greenhill, C.: Random walks on combinatorial objects. In: Lamb, J., Preece, D. (eds.) Surveys in Combinatorics, pp. 101–136. Cambridge University Press, Cambridge (1999)Google Scholar
  9. 9.
    Dyer, M., Sinclair, A., Vigoda, E., Weitz, D.: Mixing in time and space for lattice spin systems: A combinatorial view. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 149–163. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Ferreira, S., Sokal, A.: Antiferromagnetic Potts models on the square lattice: a high-precision Monte Carlo study. J. Statistical Physics 96, 461–530 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldberg, L., Martin, R., Paterson, M.: Random sampling of 3-colourings in ℤ2. Random Structures and Algorithms (to appear)Google Scholar
  12. 12.
    Jerrum, M.: A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures and Algorithms 7, 157–165 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. SIAM Computing 31, 167–192 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics, Saint-Flour 1997. Springer Lecture Notes in Mathematics, vol. 1717, pp. 93–191 (1999)Google Scholar
  15. 15.
    Moore, C., Newman, M.: Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet. J. Stat. Phys. 99, 661–690 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Moore, C., Nordahl, M., Minar, N., Shalizi, C.: Vortex dynamics and entropic forces in antiferromagnets and antiferromagnetic Potts models. Physical Review E 60, 5344–5351 (1999)CrossRefGoogle Scholar
  17. 17.
    Randall, D., Tetali, P.: Analyzing Glauber dynamics by comparison of Markov chains. J. Mathematical Physics 41, 1598–1615 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Salas, J., Sokal, A.: Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Statistical Physics 86, 551–579 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sinclair, A.: Algorithms for random generation and counting: a Markov chain approach, pp. 47–48. Birkhauser, Boston (1993)zbMATHGoogle Scholar
  20. 20.
    Sinclair, A., Jerrum, M.: Approximate counting, uniform generation, and rapidly mixing Markov chains. Information and Computation 82, 93–133 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sokal, A.: A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models. Talk presented at the conference on Inhomogeneous Random Systems, Université de Cergy-Pontoise (January 2000); Markov Processes and Related Fields 7, 21–38 (2001)Google Scholar
  22. 22.
    Vigoda, E.: Improved bounds for sampling colorings. J. Mathematical Physics 41, 1555–1569 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Wang, J., Swendsen, R., Kotecký, R.: Physical Review Letters 63, 109 (1989)CrossRefGoogle Scholar
  24. 24.
    Wang, J., Swendsen, R., Kotecký, R.: Physical Review B 42, 2465 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Dimitris Achlioptas
    • 1
  • Mike Molloy
    • 2
  • Cristopher Moore
    • 3
  • Frank Van Bussel
    • 4
  1. 1.Microsoft Research
  2. 2.Dept of Computer ScienceUniversity of Toronto, and Microsoft Research
  3. 3.Computer Science DepartmentUniversity of New Mexico
  4. 4.Dept of Computer ScienceUniversity of Toronto

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