A Dichotomy for Real Boolean Holant Problems

TL;DR

This paper establishes a clear complexity classification for Holant problems on the boolean domain with real-valued constraints, showing they are either efficiently solvable or #P-hard, based on explicit criteria.

Contribution

It provides the first comprehensive dichotomy theorem for real-valued Holant problems without symmetry or auxiliary function assumptions.

Findings

Holant problems are either P-time solvable or #P-hard.

Explicit criterion determines the complexity classification.

Involves special functions related to Bell states in quantum information.

Abstract

We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions as in previous results. It is proved that for every set $\mathcal{F}$ of real-valued constraint functions, Holant$(\mathcal{F})$ is either P-time computable or #P-hard. The classification has an explicit criterion. This is the culmination of much research on this problem, and it uses previous results and techniques from many researchers. Some particularly intriguing concrete functions $f_6$, $f_8$ and their associated families with extraordinary closure properties related to Bell states in quantum information theory play an important role in this proof.