Advertisement

Analyzing Glauber dynamics by comparison of Markov chains

  • Dana Randall
  • Prasad Tetali
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)

Abstract

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or “Glauber dynamics.” Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems.

In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating tilings, colorings and independent sets.

Keywords

Markov Chain Comparison Theorem Triangular Lattice Dirichlet Form Markov Chain Monte Carlo Algorithm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aldous, D. Random walks on finite groups and rapidly mixing Markov chains. Séminaire de Probabilités XVII, 1981/82, Springer Lecture Notes in Mathematics 986, pp. 243–297.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baxter, R.J. Exactly solved models in statistical mechanics. Academic Press, London, 1982.zbMATHGoogle Scholar
  3. 3.
    van den Berg, J. and Steif, J.E. Percolation and the hard-core lattice gas model. Stochastic Processes and their Applications 49, 1994, pp. 179–197.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Diaconis, P. and Stroock, D. Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probability 1, 1991, pp. 36–61.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Diaconis, P. and Saloff-Coste, L. Comparison theorems for reversible Markov chains. Ann. Appl. Probability 3, 1993, pp. 696–730.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Diaconis, P. and Saloff-Coste, L. Logarithmic Sobolev Inequalities for Finite Markov Chains. Ann. of Appl. Probab. 6 (1996), pp. 695–750.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Jerrum, M. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures and Algorithms 7, 1995, pp. 157–165.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Jerrum, M.R. and Sinclair, A.J. Approximating the permanent. SIAM Journal on Computing 18 (1989), pp. 1149–1178.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Jerrum, M., Valiant, L. and Vazirani, V. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science 43 (1986), pp. 169–188.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Kannan, R., Tetali, and P., Vempala, S. Simple Markov chain algorithms for generating bipartite graphs and tournaments. Proc. of the 8th ACM-SIAM Symp. on Discrete Algorithms January 1997.Google Scholar
  11. 11.
    Lieb, E.H. Residual entropy of square ice. Physical Review 162, 1967, pp. 162–172.CrossRefGoogle Scholar
  12. 12.
    Lubin, M. and Sokal A.D. Comment on “Antiferromagnetic Potts Model. Phys. Rev. Lett. 71, 1993, pp. 17–78.CrossRefGoogle Scholar
  13. 13.
    Luby, M., Randall, D. and Sinclair, A. Markov Chain Algorithms for Planar Lattice Structures. Proc. 36th IEEE Symposium on Foundations of Computing (1995), pp. 150–159.Google Scholar
  14. 14.
    Luby, M., Vigoda, E. Approximately counting up to four. Proc. 29th ACM Symposium on Theory of Computing (1997), pp. 150–159.Google Scholar
  15. 15.
    Madras, N. and Randall, D. Factoring Graphs to Bound Mixing Time. Proc. 37th IEEE Symposium on Foundations of Computing (1996).Google Scholar
  16. 16.
    Sinclair, A.J. and Jerrum, M.R. Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Computation 82 (1989), pp. 93–133.CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Sinclair, A.J. Algorithms for random generation & counting: a Markov chain approach. Birkhäuser, Boston, 1993, pp. 47–48.zbMATHGoogle Scholar
  18. 18.
    Sinclair, A.J. Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability, & Computing. 1 (1992), pp. 351–370.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Thurston, W. Conway's tiling groups. American Mathematical Monthly 97, 1990, pp. 757–773.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Vigoda, E. personal communication.Google Scholar
  21. 21.
    Welsh, D.J.A. The computational complexity of some classical problems from statistical physics. In Disorder in Physical Systems, (G. Grimmett and D. Welsh eds.). Claredon Press, Oxford, 1990, pp. 323–335.Google Scholar
  22. 22.
    Welsh, D.J.A. Approximate counting. In Surveys in Combinatorics, (R.A. Bailey, ed.). Cambridge University Press, London Math Society Lecture Notes 241, 1997, pp. 287–317.Google Scholar
  23. 23.
    Wilson, D.B. Mixing times of lozenge tiling and card shuffling Markov chains, draft of manuscript.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Dana Randall
    • 1
  • Prasad Tetali
    • 2
  1. 1.School of Mathematics and College of ComputingGeorgia Institute of TechnologyAtlantaUSA
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations