Bipartite 3-Regular Counting Problems with Mixed Signs

TL;DR

This paper establishes a clear complexity classification for bipartite 3-regular Holant problems with integer-valued symmetric functions, revealing new phenomena in planar versus general case tractability.

Contribution

It provides a complexity dichotomy for a broad class of counting problems with mixed signs and introduces a novel phenomenon of planar tractability via holographic transformations.

Findings

Problems are either P-time solvable or #P-hard based on explicit criteria.

Discovered a set of functions where planar problems are tractable but general problems are #P-hard.

Identified a new phenomenon involving holographic transformations and global arguments.

Abstract

We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of the form $\operatorname{Holant}\left(f\mid =_3 \right)$, where $f$ is any integer-valued ternary symmetric constraint function on Boolean variables, we prove that it is either P-time computable or #P-hard, depending on an explicit criterion of $f$. The constraint function can take both positive and negative values, allowing for cancellations. The dichotomy extends easily to rational valued functions of the same type. In addition, we discover a new phenomenon: there is a set $\mathcal{F}$ with the property that for every $f \in \mathcal{F}$ the problem $\operatorname{Holant}\left(f\mid =_3 \right)$ is planar P-time computable but #P-hard in general, yet its planar tractability is by a combination of a holographic transformation by $\left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right]$ to FKT together with an independent global argument.