Bipartite 3-regular counting problems with mixed signs

Abstract

We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. These are also counting CSP problems where every constraint has arity 3 and every variable is read-thrice. For every problem of the form Holant(f|=3), where f is any integer (or equivalently, rational)-valued ternary symmetric constraint function on Boolean variables, we prove that it is either P-time computable or #P-hard, depending on an explicit criterion of f. The constraint function can take both positive and negative values, allowing for cancellations. In addition, we discover a new phenomenon: there is a set F with the property that for every f∈F the problem Holant(f|=3) is planar P-time computable but #P-hard in general, yet its planar tractability is by a combination of a holographic transformation by [111−1] to FKT together with an independent global argument.