The Complexity of Partition Functions

  • Andrei Bulatov
  • Martin Grohe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)


We give a complexity theoretic classification of the counting versions of so-called H-colouring problems for graphs H that may have multiple edges between the same pair of vertices. More generally, we study the problem of computing a weighted sum of homomorphisms to a weighted graph H.

The problem has two interesting alternative formulations: First, it is equivalent to computing the partition function of a spin system as studied in statistical physics. And second, it is equivalent to counting the solutions to a constraint satisfaction problem whose constraint language consists of two equivalence relations.

In a nutshell, our result says that the problem is in polynomial time if the adjacency matrix of H has row rank 1, and #P-complete otherwise.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andrei Bulatov
    • 1
  • Martin Grohe
    • 2
  1. 1.Computing LaboratoryUniversity of OxfordOxfordUK
  2. 2.Institut für InformatikHumboldt-UniversitätBerlinGermany

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