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A Complexity Dichotomy for Partition Functions with Mixed Signs Goldberg, Leslie Ann Grohe, Martin Jerrum, Mark Thurley, Marc Computational complexity \emph{Partition functions}, also known as \emph{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 (2000) and Bulatov and Grohe (2005), 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 $\ensuremath{\mathbb{F}_2}$. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Leslie Ann Goldberg and Martin Grohe and Mark Jerrum and Marc Thurley 2009 Is Part Of LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2009.1821 urn:nbn:de:0030-drops-18217 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1821 eng https://creativecommons.org/licenses/by-nd/3.0/legalcode