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arXiv:2609.36072

Spectral Methods for the Complexity of Planar Graph Homomorphisms

Ashwin Maran, Jin-Yi Cai, Zhuxiao Tang

Published Sep 28, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2609.36072

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We explore the frontier beyond the recently discovered barrier represented by the \emph{quantum automorphism group} qut(M)qut(M)qut(M) in the classification theory of planar graph homomorphisms PlGH(M)PlGH(M)PlGH(M). We show that analyzing the spectral relations of MMM can prove #P-hardness when traditional vertex separation and domain-reduction methods with planar edge gadgets provably fail due to the \qut(M)\qut(M)\qut(M) barrier. We prove two criteria of #P-hardness for PlGH(M)PlGH(M)PlGH(M): a spectral criterion and a determinant criterion. It is known that the core problem for the classification of PlGH(M)PlGH(M)PlGH(M) for nonnegative matrices MMM is for positive definite entry-wise positive matrices. We use the spectral criterion to show that PlGH(M)PlGH(M)PlGH(M) is #P-hard for all circulant matrices of prime order q≥3q \ge 3q≥3, while for q=2q=2q=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\PlGH\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)PlGH(M)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)PlGH(M)PlGH(M) in (2) consists of precisely those computable by FKT with a holographic transformation.
We study the complexity of counting (weighted) planar graph homomorphism problem Pl-GH(M)\tt{Pl\text{-}GH}(M)Pl-GH(M) parametrized by an arbitrary symmetric non-negative real valued matrix MMM. For matrices with pairwise distinct diagonal values, we prove a complete dichotomy theorem: Pl-GH(M)\tt{Pl\text{-}GH}(M)Pl-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 MMM 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 Mančinska and Roberson to establish links between \textit{planar} edge gadgets and the theory of the \textit{quantum automorphism group} Qut(M)\tt{Qut}(M)Qut(M). We show that planar edge gadgets that can separate vertex pairs of MMM exist precisely when Qut(M)\tt{Qut}(M)Qut(M) is \emph{trivial}, and prove that the problem of whether Qut(M)\tt{Qut}(M)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.
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.
Valiant introduced match gate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, match gate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but ractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are \#P-hard on general graphs but ractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in the statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs and can be arbitrary real-valued symmetric functions. We prove that every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but tractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.
We prove a complexity classification theorem that divides the six-vertex model on graphs into exactly three types. For every setting of the parameters of the model, the computation of the partition function is precisely: Either (1) solvable in polynomial time for every graph, or (2) #P-hard for general graphs but solvable in polynomial time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. In addition to matchgates and matchgates-transformable signatures, we discover previously unknown families of planar-tractable partition functions by a non-local connection to #CSP, defined in terms of a loop space. For the proof of #P-hardness, we introduce the use of M\obius transformations as a powerful new tool to prove that certain complexity reductions succeed in polynomial time.