Dichotomy for Holant problems of Boolean domain

Abstract

Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant*(F), where F is a set of constraint functions on Boolean variables and output complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #P-hard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose inner product with itself is zero.