Spectral Methods for the Complexity of Planar Graph Homomorphisms

2026-09-28

AI Summary

This study addresses the complexity barriers introduced by quantum automorphism groups in the classification of planar graph homomorphisms. By integrating spectral theory, determinant criteria, holographic transformations, and the FKT algorithm, this work establishes a universal complexity framework categorizing problems as P, FKT-solvable, or #P-hard. Specifically, it proves #P-hardness through spectral relation analysis and proposes novel spectral and determinant criteria to bridge theoretical gaps left by conventional methods. The primary contributions include completing complexity dichotomy classifications for specific matrix classes, rigorously proving the #P-hardness of cyclic matrices of prime order, and achieving a full classification for problems involving 2×2 tensor product matrices.

Abstract

We explore the frontier beyond the recently discovered barrier represented by the \emph{quantum automorphism group} $qut(M)$ in the classification theory of planar graph homomorphisms $PlGH(M)$. We show that analyzing the spectral relations of $M$ can prove \#P-hardness when traditional vertex separation and domain-reduction methods with planar edge gadgets provably fail due to the $\qut(M)$ barrier. We prove two criteria of \#P-hardness for $PlGH(M)$: a spectral criterion and a determinant criterion. It is known that the core problem for the classification of $PlGH(M)$ for nonnegative matrices $M$ is for positive definite entry-wise positive matrices. We use the spectral criterion to show that $PlGH(M)$ is \#P-hard for all circulant matrices of prime order $q \ge 3$, while for $q=2$ it is precisely the matchgate case and is P-time computable by the FKT algorithm (for planar perfect matching). We also prove a complexity dichotomy for $\PlGH$ problems defined by tensor products of 2 by 2 matrices. This gives a complete complexity classification for this class of matrices, and the FKT algorithm together with a holographic transformation is \emph{universal}---every $PlGH(M)$ is either (1) P-time computable over all graphs, or (2) \#P-hard in general but P-time computable over planar graphs, or (3) \#P-hard over planar graphs; furthermore, $PlGH(M)$ in (2) consists of precisely those computable by FKT with a holographic transformation.

Problem

• Planar Graph Homomorphisms
• Computational Complexity
• Quantum Automorphism Group
• #P-hardness
• Complexity Dichotomy

Innovation

• Planar Graph Homomorphisms
• Spectral Methods
• Quantum Automorphism Group
• Complexity Dichotomy
• Holographic Transformation