Journal
THEORETICAL COMPUTER SCIENCE
Volume 348, Issue 2-3, Pages 148-186Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.tcs.2005.09.011
Keywords
counting complexity; partition function; graph homomorphism; constraint satisfaction
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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. (c) 2005 Elsevier B.V. All rights reserved.
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