Spin Systems on Graphs with Complex Edge Functions and Specified Degree Regularities

Abstract

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.