We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.

arXiv:2011.04827Leibniz international proceedings in informaticsNov 9, 20201 citation

Spectral Methods for the Complexity of Planar Graph Homomorphisms

Ashwin Maran, Jin-Yi Cai, Zhuxiao Tang

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.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.

arXiv:2609.36072Sep 28, 20260 citations

Universality of Graph Homomorphism Games and the Quantum Coloring Problem

Samuel J. Harris

arXiv:2305.18116Annales Henri PoincaréFeb 28, 20247 citations

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$ The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

Yuan Huang, Zhiguo Fu

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

arXiv:2609.11175Sep 10, 20260 citations

On decomposing finite abelian groups

Sándor Szabó

Acta Mathematica Academiae Scientiarum HungaricaeMar 1, 19808 citations

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.

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of

SL(2,\mathbb{C})

\mathrm{SL}(2,\mathbb{C})

and stable subgroup sequences into the proof. These new techniques are of independent interest.