A Complexity Trichotomy for K-Regular Asymmetric Spin Systems Using Number Theory.

Suppose φ and ψ are two angles satisfying tan ( φ ) = 2 tan ( ψ ) > 0 . We prove that under this condition φ and ψ cannot be both rational multiples of π. We use this number theoretic result to prove a classification of the computational complexity of spin systems on k -regular graphs with general (not necessarily symmetric) real valued edge weights. We establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: (1) Polynomial time computable, (2) #P-hard in general but polynomial time computable on planar graphs, and (3) #P-hard on planar graphs. In particular, problems in (2) are precisely those that can be transformed by a holographic reduction to a form solvable by the Fisher-Kasteleyn-Temperley algorithm for counting perfect matchings in a planar graph.