Restricted Holant Dichotomy on Domains 3 and 4

TL;DR

This paper establishes complexity dichotomies for Holant problems with arity-3 functions over domains of size 3 and 4, classifying problems as either tractable or #P-hard.

Contribution

It provides the first complete complexity classifications for Holant problems with arity-3 functions on these specific domains.

Findings

Dichotomy theorem for domain size 3 functions

Dichotomy theorem for domain size 4 functions

Clear boundary between tractable and #P-hard cases

Abstract

$\operatorname{Holant}^*(f)$ denotes a class of counting problems specified by a constraint function $f$. We prove complexity dichotomy theorems for $\operatorname{Holant}^*(f)$ in two settings: (1) $f$ is any arity-3 real-valued function on input of domain size 3. (2) $f$ is any arity-3 $\{0,1\}$-valued function on input of domain size 4.