The computational complexity of Holant problems on 3-regular graphs

Abstract

Holant problem is a framework to study counting problems, which is expressive enough to contain Counting Graph Homomorphisms (#GH) and Counting Constraint Satisfaction Problems (#CSP) as special cases. In the present paper, we classify the computational complexity of Holant problems on 3-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we prove that Holant problem on 3-regular graphs is #P-hard except for the signature is not genuinely entangled, A-transformable, P-transformable or vanishing, in which cases the problem is tractable.