Dichotomy for Real Holantc Problems

Abstract

Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic 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 intricate polynomial time tractable problems in the broader framework. We discover a new family of constraint functions 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 pinning functions. Previously, dichotomy for the same framework 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 technical component toward this Holant dichotomy.