Planar Graph Homomorphisms: A Dichotomy and a Barrier from Quantum Groups

2026-01-30

AI Summary

This study investigates the computational complexity of counting homomorphisms from planar graphs to a fixed target, denoted $\text{Pl-GH}(M)$, where the target is parameterized by a nonnegative real symmetric matrix $M$. For matrices $M$ with pairwise distinct diagonal entries, the paper establishes the first complete dichotomy theorem: $\text{Pl-GH}(M)$ is either solvable in polynomial time or $\#\mathsf{P}$-hard. The key insight is an exact correspondence between the expressiveness of planar edge gadgets and the triviality of the quantum automorphism group $\text{Qut}(M)$. Moreover, the authors prove that determining this triviality is undecidable, thereby identifying a fundamental barrier to complexity classification in the planar setting. By integrating graph homomorphism theory, planar gadget constructions, representation theory of quantum groups, and computability theory, this work precisely delineates the boundary of applicability for such dichotomy results.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the complexity of counting (weighted) planar graph homomorphism problem $\tt{Pl\text{-}GH}(M)$ parametrized by an arbitrary symmetric non-negative real valued matrix $M$. For matrices with pairwise distinct diagonal values, we prove a complete dichotomy theorem: $\tt{Pl\text{-}GH}(M)$ is either polynomial-time tractable, or $\#$P-hard, according to a simple criterion. More generally, we obtain a dichotomy whenever every vertex pair of the graph represented by $M$ can be separated using some planar edge gadget. A key question in proving complexity dichotomies in the planar setting is the expressive power of planar edge gadgets. We build on the framework of Man\v{c}inska and Roberson to establish links between \textit{planar} edge gadgets and the theory of the \textit{quantum automorphism group} $\tt{Qut}(M)$. We show that planar edge gadgets that can separate vertex pairs of $M$ exist precisely when $\tt{Qut}(M)$ is \emph{trivial}, and prove that the problem of whether $\tt{Qut}(M)$ is trivial is undecidable. These results delineate the frontier for planar homomorphism counting problems and uncover intrinsic barriers to extending nonplanar reduction techniques to the planar setting.

Problem

• planar graph homomorphism
• computational complexity
• dichotomy
• quantum automorphism group
• counting problem

Innovation

• planar graph homomorphism
• complexity dichotomy
• quantum automorphism group
• planar edge gadget
• undecidability