Dichotomy Result on 3-Regular Bipartite Non-negative Functions

TL;DR

This paper establishes a clear complexity classification for a class of Holant problems on 3-regular bipartite graphs with nonnegative symmetric functions, identifying conditions under which they are polynomial-time solvable or -hard.

Contribution

It proves an explicit dichotomy theorem for Holant problems with nonnegative symmetric functions on 3-regular bipartite graphs, determining their computational complexity.

Findings

Holant problems are either polynomial-time solvable or -hard based on explicit criteria.

The dichotomy applies to functions with nonnegative symmetric weights on 3-regular bipartite graphs.

The paper provides a complete classification for this class of problems.

Abstract

We prove a complexity dichotomy theorem for a class of Holant problems on 3-regular bipartite graphs. Given an arbitrary nonnegative weighted symmetric constraint function $f = [x_0, x_1, x_2, x_3]$, we prove that the bipartite Holant problem $\operatorname{Holant} \left( f \mid \left( =_3 \right) \right)$ is \emph{either} computable in polynomial time \emph{or} $\#$P-hard. The dichotomy criterion on $f$ is explicit.