发表机构
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出谱判据和行列式判据，证明平面图同态问题在循环矩阵和张量积矩阵下的复杂性分类，并展示FKT算法与全息变换的普适性。
AI 中文摘要
我们探索了在平面图同态 $PlGH(M)$ 的分类理论中，由量子自同构群 $qut(M)$ 所代表的新近发现的障碍之外的前沿。我们表明，分析 $M$ 的谱关系可以证明 \u0023P-困难性，而传统的顶点分离和带平面边小工具的域约简方法由于 $qut(M)$ 障碍而可证明地失败。我们证明了 $PlGH(M)$ 的 \u0023P-困难性的两个判据：谱判据和行列式判据。已知非负矩阵 $M$ 的 $PlGH(M)$ 分类的核心问题是针对逐项正的正定矩阵。我们使用谱判据来证明对于所有素数阶 $q \ge 3$ 的循环矩阵，$PlGH(M)$ 是 \u0023P-困难的，而对于 $q=2$，它恰好是匹配门情形，并且可由 FKT 算法（用于平面完美匹配）在 P 时间内计算。我们还证明了由 2×2 矩阵的张量积定义的 $\PlGH$ 问题的复杂性二分法。这给出了此类矩阵的完整复杂性分类，并且 FKT 算法连同全息变换是通用的——每个 $PlGH(M)$ 要么 (1) 在所有图上 P 时间可计算，要么 (2) 在一般情况下是 \u0023P-困难的但在平面图上 P 时间可计算，要么 (3) 在平面图上是 \u0023P-困难的；此外，(2) 中的 $PlGH(M)$ 恰好是由 FKT 算法连同全息变换可计算的那些。
英文摘要
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.
Comments35 pages, 2 figures
