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Rapid Mixing of Subset Glauber Dynamics on Graphs of Bounded Tree-Width

  • Magnus Bordewich
  • Ross J. Kang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

Motivated by the ‘subgraphs world’ view of the ferromagnetic Ising model, we develop a general approach to studying mixing times of Glauber dynamics based on subset expansion expressions for a class of graph polynomials. With a canonical paths argument, we demonstrate that the chains defined within this framework mix rapidly upon graphs of bounded tree-width. This extends known results on rapid mixing for the Tutte polynomial, the adjacency-rank (R 2-)polynomial and the interlace polynomial.

Keywords

Markov chain Monte Carlo graph polynomials subset expansion tree-width canonical paths randomised approximation schemes rapid mixing 

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References

  1. 1.
    Alon, N., Frieze, A., Welsh, D.: Polynomial time randomized approximation schemes for Tutte-Gröthendieck invariants: the dense case. Random Structures Algorithms 6(4), 459–478 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrzejak, A.: An algorithm for the Tutte polynomials of graphs of bounded treewidth. Discrete Math. 190(1-3), 39–54 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arratia, R., Bollobás, B., Sorkin, G.B.: A two-variable interlace polynomial. Combinatorica 24(4), 567–584 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields 131(3), 311–340 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birkhoff, G.D.: A determinant formula for the number of ways of coloring a map. Ann. of Math. (2) 14(1-4), 42–46 (1912/1913)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bläser, M., Hoffmann, C.: Fast evaluation of interlace polynomials on graphs of bounded treewidth. To appear in Algorithmica, doi:10.1007/s00453-010-9439-4Google Scholar
  7. 7.
    Bläser, M., Hoffmann, C.: On the complexity of the interlace polynomial. In: Albers, S., Weil, P. (eds.) 25th International Symposium on Theoretical Aspects of Computer Science (STACS 2008), Dagstuhl, Germany. Leibniz International Proceedings in Informatics (LIPIcs), vol. 1, pp. 97–108. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2008)Google Scholar
  8. 8.
    Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Bordewich, M.: Approximating the number of acyclic orientations for a class of sparse graphs. Combin. Probab. Comput. 13(1), 1–16 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Courcelle, B.: A multivariate interlace polynomial and its computation for graphs of bounded clique-width. Electron. J. Combin. 15(1): Research Paper 69, 36 (2008)Google Scholar
  11. 11.
    Dembo, A., Montanari, A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2), 565–592 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ding, J., Lubetzky, E., Peres, Y.: Mixing time of critical Ising model on trees is polynomial in the height. Comm. Math. Phys. 295(1), 161–207 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ellis-Monaghan, J.A., Merino, C.: Graph polynomials and their applications I: The Tutte polynomial. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 219–255. Birkhäuser, Boston (2011)CrossRefGoogle Scholar
  14. 14.
    Ellis-Monaghan, J.A., Merino, C.: Graph polynomials and their applications II: Interrelations and interpretations. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 257–292. Birkhäuser, Boston (2011)CrossRefGoogle Scholar
  15. 15.
    Fomin, F.V., Thilikos, D.M.: A 3-approximation for the pathwidth of Halin graphs. J. Discrete Algorithms 4(4), 499–510 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ge, Q., Štefankovič, D.: A graph polynomial for independent sets of bipartite graphs. CoRR, abs/0911.4732 (2009)Google Scholar
  17. 17.
    Glauber, R.J.: Time-dependent statistics of the Ising model. J. Mathematical Phys. 4, 294–307 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Goldberg, L.A., Jerrum, M.: Personal communication (2010)Google Scholar
  19. 19.
    Goldberg, L.A., Jerrum, M.: Approximating the partition function of the ferromagnetic potts model. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 396–407. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Goldberg, L.A., Jerrum, M., Karpinski, M.: The mixing time of Glauber dynamics for coloring regular trees. Random Structures Algorithms 36(4), 464–476 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Grimmett, G.: The random-cluster model. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 333. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hliněný, P., Oum, S.-i., Seese, D., Gottlob, G.: Width Parameters Beyond Tree-width and their Applications. The Computer Journal 51(3), 326–362 (2008)CrossRefGoogle Scholar
  24. 24.
    Ising, E.: Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei 31, 253–258 (1925), doi:10.1007/BF02980577Google Scholar
  25. 25.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108(1), 35–53 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jerrum, M.: Counting, sampling and integrating: algorithms and complexity, ETH Zürich. Lectures in Mathematics. Birkhäuser, Basel (2003)CrossRefzbMATHGoogle Scholar
  27. 27.
    Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22(5), 1087–1116 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc (N.S.) 12(1), 103–111 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Karger, D.R.: A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM Rev. 43(3), 499–522 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Inform. Process. Lett. 42(6), 345–350 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Makowsky, J.A.: From a zoo to a zoology: towards a general theory of graph polynomials. Theory Comput. Syst. 43(3-4), 542–562 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Makowsky, J.A., Mariño, J.P.: Farrell polynomials on graphs of bounded tree width. Adv. in Appl. Math. 30(1-2), 160–176 (2003), Formal power series and algebraic combinatorics, Scottsdale, AZ (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, colorings, and other models on trees. Random Structures Algorithms 31(2), 134–172 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. The Journal of Chemical Physics 21(6), 1087–1092 (1953)CrossRefGoogle Scholar
  35. 35.
    Noble, S.D.: Evaluating the Tutte polynomial for graphs of bounded tree-width. Combin. Probab. Comput. 7(3), 307–321 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Noble, S.D.: Evaluating a weighted graph polynomial for graphs of bounded tree-width. Electron. J. Combin. 16(1): research Paper 64, 14 (2009)Google Scholar
  37. 37.
    Noble, S.D., Welsh, D.J.A.: A weighted graph polynomial from chromatic invariants of knots. Ann. Inst. Fourier 49(3), 1057–1087 (1999), Symposium à la Mémoire de François Jaeger, Grenoble (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Potts, R.B.: Some generalized order-disorder transformations. Proc. Cambridge Philos. Soc. 48, 106–109 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Randall, D.: Rapidly mixing Markov chains with applications in computer science and physics. Computing in Science Engineering 8(2), 30–41 (2006)CrossRefGoogle Scholar
  40. 40.
    Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Surveys in Combinatorics 2005. London Math. Soc. Lecture Note Ser., vol. 327, pp. 173–226. Cambridge Univ. Press, Cambridge (2005)CrossRefGoogle Scholar
  41. 41.
    Tetali, P., Vera, J.C., Vigoda, E., Yang, L.: Phase transition for the mixing time of the Glauber dynamics for coloring regular trees. In: Charikar, M. (ed.) SODA, pp. 1646–1656. SIAM, Philadelphia (2010)Google Scholar
  42. 42.
    Thomas, R.: Tree-decompositions of graphs (1996), Lecture notes http://www.math.gatech.edu/~thomas/tree.ps
  43. 43.
    Tutte, W.T.: A contribution to the theory of chromatic polynomials. Canadian J. Math. 6, 80–91 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Welsh, D.J.A.: Complexity: knots, colourings and counting. London Mathematical Society Lecture Note Series, vol. 186. Cambridge University Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Magnus Bordewich
    • 1
  • Ross J. Kang
    • 1
  1. 1.Durham UniversityDurhamUK

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