Dichotomy for Holant∗ Problems on the 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 $\text {Holant}^{*}(\mathcal {F})$ , where ${\mathcal {F}}$ is a set of constraint functions on Boolean variables and taking 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 with the broader scope that allows complex-valued constraint functions, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose dot product with itself is zero.