Bipartite 3-Regular Counting Problems with Mixed Signs

We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of the form Holant (f vertical bar= 3 ), where f is any integer-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 is an element of F the problem Holant (f vertical bar= 3 ) is planar P-time computable but #P-hard in general, yet its planar tractability is by a combination of a holographic transformation by [GRAPHICS] to FKT together with an independent global argument.