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hal-00466411, version 3

A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Andrea Bedini () 1, Jesper Jacobsen () 2

Résumé : Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.

  • 1 :  Istituto Nazionale di Fisica Nucleare, Sezione di Milano (INFN, Sezione di Milano)
  • INFN
  • 2 :  Laboratoire de Physique Théorique de l'ENS (LPTENS)
  • CNRS : UMR8549 – Université Pierre et Marie Curie (UPMC) - Paris VI – Ecole normale supérieure de Paris - ENS Paris
  • Domaine : Physique/Physique mathématique
    Mathématiques/Physique mathématique
    Physique/Matière Condensée/Mécanique statistique
    Mathématiques/Combinatoire
    Informatique/Complexité
  • Mots-clés : computational complexity – transfer matrix – tree decomposition – chromatic polynomial
  • Commentaire : 5 pages – 3 figures. Version 2 has been substantially expanded. Version 3 shows that the worst-case running time is sub-exponential in the number of vertices.
  • Versions disponibles :  v1 (25-03-2010) v2 (31-05-2010) v3 (06-08-2010)
 
  • hal-00466411, version 3
  • http://hal.archives-ouvertes.fr/hal-00466411
  • oai:hal.archives-ouvertes.fr:hal-00466411
  • Contributeur : Jesper Jacobsen <>
  • Soumis le : Vendredi 6 Août 2010, 15:30:01
  • Dernière modification le : Vendredi 6 Août 2010, 16:03:02