Dichotomy for real holantc problems

Abstract

Holant problems capture a class of Sum-of-Product compu-tations such as counting matchings. It is inspired by holo-graphic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A complexity classification for Holant problems is more difficult to prove, not only because it logically implies a classification for counting CSP, but also due to the deeper reason that there exist more in-tricate polynomial time tractable problems in the broader framework. We discover a new family of constraint func-tions L which define polynomial time computable counting problems. These do not appear in counting CSP, and no newly discovered tractable constraints can be symmetric. It has a delicate support structure related to error-correcting codes. Local holographic transformations is fundamental in its tractability. We prove a complexity dichotomy theorem for all Holant problems defined by any real valued constraint function set on Boolean variables and contains two 0-1 pin-ning functions. Previously, dichotomy for the same frame-work was only known for symmetric constraint functions. The set L supplies the last piece of tractability. We also prove a dichotomy for a variant of counting CSP as a tech-nical component toward this Holant dichotomy.