From an odd arity signature to a Holant dichotomy

TL;DR

This paper establishes a complexity dichotomy for complex-valued Holant problems on the Boolean domain with odd arity signatures, classifying problems as either efficiently solvable or #P-hard, and introduces a generalized decomposition lemma for these signatures.

Contribution

It proves a new complexity dichotomy for complex-valued Holant problems with odd arity signatures and generalizes the decomposition lemma for these signatures.

Findings

Dichotomy classifies problems as FP^NP or #P-hard.

Generalized decomposition lemma for complex-valued Holant signatures.

Method aids in constructing reductions for complex-valued Holant problems.

Abstract

\textsf{Holant} is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued \textsf{Holant} on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued \textsf{Holant} on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for \textsf{\#EO}, and consequently is an $\text{FP}^\text{NP}$ vs. \#P dichotomy as well, stating that each problem is either in $\text{FP}^\text{NP}$ or \#P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued \textsf{Holant} on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in $\text{FP}^\text{NP}$. We believe that this result is a powerful method for building reductions in complex-valued \textsf{Holant}, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.