New Planar Algorithms and a Full Complexity Classification of the Eight-Vertex Model

2026-02-11

AI Summary

This work fully characterizes the computational complexity of the partition function of the planar eight-vertex model for arbitrary complex parameters. By employing combinatorial transformations and holographic algorithms, the model is reduced to planar even-coloring problems and solvable six-vertex cases. Integrating tools from complex analysis, algebraic number theory, and cyclotomic field theory, the study establishes three precise complexity classes: polynomial-time solvable (P), tractable only on planar graphs (planar-P) yet #P-hard in general, and #P-hard even when restricted to planar graphs. Beyond identifying a new family of solvable models that transcend the scope of Kasteleyn’s algorithm, this work uncovers profound non-local connections to the bipartite Ising model, conformal lattice interpolation, and Möbius transformations.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We prove a complete complexity classification theorem for the planar eight-vertex model. For every parameter setting in ${\mathbb C}$ for the eight-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. In (2), we discover new P-time computable eight-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. They are obtained by a combinatorial transformation to the planar {\sc Even Coloring} problem followed by a holographic transformation to the tractable cases in the planar six-vertex model. In the process, we also encounter non-local connections between the planar eight vertex model and the bipartite Ising model, conformal lattice interpolation and M\"{o}bius transformation from complex analysis. The proof also makes use of cyclotomic fields.

Problem

• eight-vertex model
• complexity classification
• planar graphs
• partition function
• #P-hard

Innovation

• eight-vertex model
• complexity classification
• holographic transformation
• planar graph
• cyclotomic fields