Spin systems on k-regular graphs with complex edge functions

摘要

Let k ≥ 1 be an integer and let h = [{under}(h(0,0)) {down}(h(1,0)) {under}(h(0,1)){down}(h(1,1))] 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 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 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.