Spin systems on k-regular graphs with complex edge functions

Let k≥1 be an integer and let be a complex-valued symmetric function on domain {0,1} (i.e., where h(0,1)=h(1,0)). 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 the function h is attached to each edge, compute . is known as the partition function of the spin system, also known as counting 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.