A Complexity Dichotomy for Partition Functions with Mixed Signs

Abstract : Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [7] and Bulatov and Grohe [6], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices — these turn out to be central in our proofs — we obtain a simple algebraic tractability criterion, which says that the tractable cases are those “representable” by a quadratic polynomial over the field F2.
Type de document :
Communication dans un congrès
Susanne Albers and Jean-Yves Marion. 26th International Symposium on Theoretical Aspects of Computer Science STACS 2009, Feb 2009, Freiburg, Germany. IBFI Schloss Dagstuhl, pp.493-504, 2009, Proceedings of the 26th Annual Symposium on the Theoretical Aspects of Computer Science
Liste complète des métadonnées

Littérature citée [20 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/inria-00359820
Contributeur : Publications Loria <>
Soumis le : lundi 9 février 2009 - 14:35:23
Dernière modification le : mercredi 24 janvier 2018 - 11:58:01
Document(s) archivé(s) le : mardi 8 juin 2010 - 22:07:13

Fichier

41-goldberg.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : inria-00359820, version 1

Collections

Citation

Leslie Ann Goldberg, Martin Grohe, Mark Jerrum, Marc Thurley. A Complexity Dichotomy for Partition Functions with Mixed Signs. Susanne Albers and Jean-Yves Marion. 26th International Symposium on Theoretical Aspects of Computer Science STACS 2009, Feb 2009, Freiburg, Germany. IBFI Schloss Dagstuhl, pp.493-504, 2009, Proceedings of the 26th Annual Symposium on the Theoretical Aspects of Computer Science. 〈inria-00359820〉

Partager

Métriques

Consultations de la notice

71

Téléchargements de fichiers

98