8 A complexity dichotomy for partition functions with mixed signs ∗

Partition functions, also known as homomorphism functions, form a richfamily of graph invariants that contain combinatorial invariants such as thenumber of k-colourings or the number of independent sets of a graph and alsothe partition functions of certain “spin glass” models of statistical physicssuch 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 functionis either computable in polynomial time or #P-complete. Partition functionsare described by symmetric matrices with real entries, and we prove that it isdecidable in polynomial time in terms of the matrix whether a given partitionfunction is in polynomial time or #P-complete.While in general it is very complicated to give an explicit algebraic or com-binatorial description of the tractable cases, for partition functions describedby a Hadamard matrices — these turn out to be central in our proofs — weobtain a simple algebraic tractability criterion, which says that the tractablecases are those “representable” by a quadratic polynomial over the field F2. Partly funded by the EPSRC grant “The complexity of counting in constraint satisfactionproblems”.Department of Computer Science, University of Liverpool, Liverpool L69 3BX, UKInstitut für Informatik, Humboldt-Universität zu Berlin, 10099 Berlin, GermanySchool of Mathematical Sciences, Queen Mary, University of London, Mile End Road, LondonE1 4NS, UKThis research was supported by the Deutsche Forschungsgemeinschaft within the researchtraining group ’Methods for Discrete Structures’ (GRK 1408)