The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and algorithms
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
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)
Towards a dichotomy theorem for the counting constraint satisfaction problem
Information and Computation
On counting homomorphisms to directed acyclic graphs
Journal of the ACM (JACM)
SIAM Journal on Computing
The Complexity of the Counting Constraint Satisfaction Problem
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
A Decidable Dichotomy Theorem on Directed Graph Homomorphisms with Non-negative Weights
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
A Complexity Dichotomy for Partition Functions with Mixed Signs
SIAM Journal on Computing
A computational proof of complexity of some restricted counting problems
Theoretical Computer Science
Non-negatively Weighted #CSP: An Effective Complexity Dichotomy
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
A dichotomy for k-regular graphs with {0, 1}-vertex assignments and real edge functions
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Gadgets and anti-gadgets leading to a complexity dichotomy
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Spin systems on k-regular graphs with complex edge functions
Theoretical Computer Science
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Let k ≥ 1 be an integer and h = [h(00) h(01) h(10) h(11)], where h(01) = h(10), be a complex-valued (symmetric) function h on domain {0, 1}. We introduce a new technique, called a syzygy, and prove a dichotomy theorem for the following class of problems, specified by k and h: Given an arbitrary k-regular graph G = (V, E), where each edge is attached the function h, compute Z(G) =Σσ:V →{0,1} Π{u,v}∈E h(σ(u), σ(v)). Z(ċ) is known as the partition function of the spin system, also known as graph homomorphisms on domain size two, and is a special case of Holant problems. The dichotomy theorem gives a complete classification of the computational complexity of this problem, depending on k and h. The dependence on k and h is explicit. We also extend this classification to graphs with deg(v), for all v ∈ V, belonging to a specified set of degrees.