A Dichotomy for Planar Graph Homomorphisms with Nonnegative Weights

2026-10-04

AI Summary

This study addresses the computational complexity classification of counting graph homomorphisms on planar graphs with fixed non-negative weights. The proposed method combines entropy continuity arguments with distance geometry to recover hidden coordinates, thereby reducing factors to zero-field Ising interactions. Through a maximum entropy completion technique, this work establishes the first explicit tractability criterion for symmetric non-negative matrices of arbitrary order, extending the framework from algebraic to real-valued weights. As a key contribution, it achieves a complete complexity dichotomy classification for systems such as clock models, with all results formally verified using Lean 4.

Abstract

We prove a complete complexity dichotomy for planar graph homomorphism counting with any fixed symmetric nonnegative matrix of arbitrary finite order, giving an explicit criterion for tractability. We also characterize exactly which fixed positive vertex weights preserve tractability, with both classifications extending from algebraic weights to fixed real weights in a prescribed exact representation. Our proof hinges on an entropy-based continuation argument: maximal logarithmic support identifies distance kernels as maximum-entropy completions, extending their positive definiteness throughout the parameter interval. This enables distance geometry to recover hidden product coordinates even when planar gadgets cannot distinguish colors; counting-hardness arguments then force the factors to be zero-field Boolean Ising interactions. The classification also yields complete tractability criteria for clock models, coupled Ising systems, and planar contractions of stoquastic imaginary-time kernels. All results have been formally verified in Lean 4.

Problem

• planar graph homomorphisms
• counting complexity
• complexity dichotomy
• nonnegative weights
• tractability criterion

Innovation

• planar graph homomorphisms
• complexity dichotomy
• entropy-based continuation
• distance geometry
• formal verification