# New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

## TL;DR

This paper classifies all P-time computable six-vertex models on planar graphs, discovering new models beyond existing algorithms and establishing a comprehensive complexity landscape with explicit criteria.

## Contribution

It introduces a complete classification of six-vertex models on planar graphs, discovering new P-time cases and providing explicit criteria for complexity.

## Key findings

- Identified all P-time computable six-vertex models on planar graphs.
- Discovered new models beyond Kasteleyn's algorithm.
- Established a comprehensive complexity classification with explicit criteria.

## Abstract

We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. The new P-time cases in (2) provably cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local connection to #CSP, defined in terms of a "loop space". This is the first substantive advance toward a planar Holant classification with not necessarily symmetric constraints. We introduce M\"obius transformation on ${\mathbb C}$ as a powerful new tool in hardness proofs for counting problems.