On the complexity of H-coloring
Journal of Combinatorial Theory Series B
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
Finite fields
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and algorithms
The complexity of choosing an H-colouring (nearly) uniformly at random
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Computational Complexity of ({\it XOR, AND\/})-Counting Problems
The Computational Complexity of ({\'it XOR, AND\'/})-Counting Problems
Towards a Dichotomy Theorem for the Counting Constraint Satisfaction Problem
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
A dichotomy theorem for constraint satisfaction problems on a 3-element set
Journal of the ACM (JACM)
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
The rank of connection matrices and the dimension of graph algebras
European Journal of Combinatorics
Inapproximability of the Tutte polynomial
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On counting homomorphisms to directed acyclic graphs
Journal of the ACM (JACM)
Graph parameters and semigroup functions
European Journal of Combinatorics
Graphs, polymorphisms and the complexity of homomorphism problems
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The Complexity of the Counting Constraint Satisfaction Problem
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
A Complexity Dichotomy For Hypergraph Partition Functions
Computational Complexity
Complexity of the cover polynomial
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Spin systems on graphs with complex edge functions and specified degree regularities
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Guest column: complexity dichotomies of counting problems
ACM SIGACT News
Gadgets and anti-gadgets leading to a complexity dichotomy
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Complexity of counting CSP with complex weights
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Spin systems on k-regular graphs with complex edge functions
Theoretical Computer Science
Lee-Yang theorems and the complexity of computing averages
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
A complete dichotomy rises from the capture of vanishing signatures: extended abstract
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Partition functions on k-regular graphs with {0,1}-vertex assignments and real edge functions
Theoretical Computer Science
The complexity of planar boolean #CSP with complex weights
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
The complexity of complex weighted Boolean #CSP
Journal of Computer and System Sciences
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Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of $k$-colorings 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 [Random Structures Algorithms, 17 (2000), pp. 260-289] and Bulatov and Grohe [Theoret. Comput. Sci., 348 (2005), pp. 148-186], 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 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 $\mathbb{F}_2$.