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

TL;DR

This paper establishes a complete complexity dichotomy for planar graph homomorphism counting problems based on matrix properties, linking the problem's complexity to quantum automorphism groups and revealing fundamental barriers in planar reductions.

Contribution

It proves a dichotomy theorem for planar graph homomorphism counting problems and connects the problem's complexity to the triviality of quantum automorphism groups, highlighting intrinsic planar barriers.

Findings

Dichotomy between polynomial-time solvable and P-hard cases based on matrix criteria.

Existence of planar edge gadgets correlates with trivial quantum automorphism groups.

Decidability of quantum automorphism group triviality is shown to be undecidable.

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.