Entrywise Logarithmic Matrix Algebra and Dichotomy of Planar Graph Homomorphisms (Part I)

2026-09-30

AI Summary

This study investigates the computational complexity classification of counting graph homomorphisms on planar graphs with non-negative weight matrices. Methodologically, it introduces an innovative element-wise logarithmic matrix algebra and integrates core techniques including holographic transformations, polynomial interpolation, and the FKT algorithm. The primary contributions are twofold: it rigorously establishes a complexity dichotomy between P and #P-hardness for positive definite and positive-entry matrices, thereby constructing a novel theoretical framework. Furthermore, this work lays a solid foundation for ultimately achieving a complete classification in the general non-negative weight matrix setting.

Abstract

We prove a complexity classification of counting planar graph homomorphisms with non-negative weights. For a real symmetric matrix $M$ with non-negative entries, the problem $\PlGH(M)$ 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 that involve the P-time FKT algorithm to count planar perfect matchings with a holographic transformation. The dichotomy is achieved by forming a (centered) logarithmic matrix algebra (a vector space with bilinear multiplication) by taking entrywise logarithms of all realizable matrices from $M$ using planar edge gadgets and polynomial interpolation. The current version is part I, which contains the proof for the dichotomy of entrywise positive and positive definite matrices, which is at the core of the dichotomy for non-negative matrices. Part II contains the extension from entrywise positive and positive definite matrices to non-negative matrices.

Problem

• planar graph homomorphisms
• complexity classification
• counting problems
• dichotomy
• #P-hard

Innovation

• Planar Graph Homomorphisms
• Logarithmic Matrix Algebra
• Complexity Dichotomy
• Holographic Transformation
• FKT Algorithm