The complexity of partition functions

  • Authors:

  • Andrei Bulatov;Martin Grohe

  • Affiliations:

  • School of Computing Science, Simon Fraser University, Burnaby, Canada;Institut für Informatik, Humboldt-Universität, Unter den Linden, Berlin, Germany

  • Venue:
  • Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
  • Year:
  • 2005

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Abstract

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-hard otherwise.