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arXiv:2601.23198

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

Jin‐Yi Cai, Ashwin Maran, Ben Young

Published Jan 30, 2026• 0 citations
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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 Mančinska 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.

DOI
10.48550/arxiv.2601.23198

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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 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 $\qut(M)$ 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 $\PlGH$ 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 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.
Mančinska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they admit the same number of homomorphisms from any planar graph. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general, which motivates the study of its relaxations. In the classical setting, Roberson and Seppelt [ICALP'23] characterized the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism in terms of equality of homomorphism counts from an appropriate graph class. The NPA hierarchy, a noncommutative generalization of the Lasserre hierarchy, provides a sequence of semidefinite programming relaxations for quantum isomorphism. In the quantum setting, we show that the feasibility of each level of the NPA hierarchy for quantum isomorphism is equivalent to equality of homomorphism counts from an appropriate class of planar graphs. Combining this characterization with the convergence of the NPA hierarchy, and noting that the union of these classes is the set of all planar graphs, we obtain a new proof of the result of Mančinska and Roberson [FOCS'20] that avoids the use of quantum groups. Moreover, this homomorphism indistinguishability characterization also yields a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.
Mančinska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they are homomorphism indistinguishable over the class of planar graphs. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general. The NPA hierarchy gives a sequence of semidefinite programming relaxations of quantum isomorphism. Recently, Roberson and Seppelt [ICALP'23] obtained a homomorphism indistinguishability characterization of the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism. We prove a quantum analogue of this result by showing that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. By combining the convergence of the NPA hierarchy with the fact that the union of these graph classes is the set of all planar graphs, we are able to give a new proof of the result of Mančinska and Roberson [FOCS'20] that avoids the use of the theory of quantum groups. This homomorphism indistinguishability characterization also allows us to give a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.
We introduce an idea called anti-gadgets for reductions in complexity theory. These anti-gadgets are presented as graph fragments, but their effect is equivalent to erasing the presence of other graph fragments, as if we had managed to include a negative copy of a certain graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z ( G ) of spin systems over 3-regular directed graphs G , Z ( G ) = ∑ σ : V ( G ) → { 0,1} ∏ ( u , v ) ∈ E ( G ) f (σ ( u ), σ ( v )), where each edge is given a (not necessarily symmetric) complex-valued binary function f : { 0,1} 2 → C. We show that Z ( G ) is either computable in polynomial time or #P-hard, depending explicitly on f . When the input graph G is planar, there is an additional class of polynomial time computable partition functions Z ( G ), while everything else remains #P-hard. Furthermore, this additional class is precisely those that can be transformed by a holographic reduction to matchgates, followed by the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.
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.