Let k ≥ 1 be an integer and h = [{under}(h(00) h(11)){down}(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) = ∑_(σ:V→{0,1}) ∏_({u,v}∈E)h(σ(u),σ(v)). Z(·) 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.
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