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

Abstract

Suppose $$\varphi$$ and $$\psi$$ are two angles satisfying $$\tan(\varphi) = 2 \tan(\psi) > 0$$ . We prove that under this condition $$\varphi$$ and $$\psi$$ 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.