A Complexity Dichotomy for Partition Functions with Mixed Signs - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2009

A Complexity Dichotomy for Partition Functions with Mixed Signs


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.
Fichier principal
Vignette du fichier
41-goldberg.pdf (261.3 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

inria-00359820 , version 1 (09-02-2009)


  • HAL Id : inria-00359820 , version 1


Leslie Ann Goldberg, Martin Grohe, Mark Jerrum, Marc Thurley. A Complexity Dichotomy for Partition Functions with Mixed Signs. 26th International Symposium on Theoretical Aspects of Computer Science STACS 2009, Feb 2009, Freiburg, Germany. pp.493-504. ⟨inria-00359820⟩


59 View
180 Download


Gmail Facebook Twitter LinkedIn More