用于判定有界团宽同态不可区分性的稠密Weisfeiler-Leman算法
A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability
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中文总结 AI 辅助
本文提出稠密Weisfeiler-Leman算法，首次判定有界团宽稠密图类上的同态不可区分性，还证明CMSO₁定义的有界团宽图类的同态不可区分性可在随机指数时间内判定，有界线性团宽图类的该问题为PSPACE完全。
中文摘要 AI 辅助
图G和H在图类F上是同态不可区分的，当且仅当它们从F中的每个图都接受相同数量的同态映射。由此衍生出多种图同构的放松情形：所有图类上的同构（Lovász，Acta Math. Hung. 1967）、树宽≤k的图上k维Weisfeiler-Leman算法下的等价性（Dvořák，J. Graph Theory 2010）、平面图上的量子同构（Mančinska-Roberson，FOCS 2020）。由于图类F通常是无限的，先验来看无法确定F上的同态不可区分性是否可判定；而平面图类的同态不可区分性是不可判定的。此前已知可判定的图类均为稀疏类。本文给出稠密图类的首个可判定性结果：提出稠密Weisfeiler-Leman算法，该算法可判定团宽≤k的图类上的同态不可区分性，团宽是树宽的稠密对应，此前该关系的可判定性未知。该算法对顶点子集的k元组着色，而非顶点的k元组。除团宽≤k的所有图类外，本文还证明了一个通用元定理：每个由CMSO₁定义的有界团宽图类上的同态不可区分性，都可在随机指数时间内判定；对于有界线性团宽的图类，时间复杂度可提升至PSPACE，且通过构造此类图类证明该问题是PSPACE完全的。这些是稠密图类上同态不可区分性的首个通用算法。
英文摘要
Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph in $\mathcal{F}$. A wide range of relaxations of graph isomorphism arise this way: isomorphism itself over the class of all graphs [Lovász, Acta Math. Hung. 1967], equivalence under the $k$-dimensional Weisfeiler-Leman algorithm over the graphs of treewidth $\leq k$ [Dvořák, J. Graph Theory 2010], and quantum isomorphism over planar graphs [Mančinska-Roberson, FOCS 2020]. Since the class $\mathcal{F}$ is typically infinite, it is not clear a priori whether homomorphism indistinguishability over $\mathcal{F}$ is decidable; for planar graphs it is undecidable. Every class for which decidability was previously known is sparse. We give the first decidability results for dense graph classes: We introduce the dense Weisfeiler-Leman algorithm that decides homomorphism indistinguishability over the class of graphs of cliquewidth $\leq k$, the dense counterpart of treewidth. This relation was not previously known to be decidable. The algorithm colors $k$-tuples of vertex subsets rather than $k$-tuples of vertices. Beyond the class of all graphs of cliquewidth $\leq k$, we prove a general meta-theorem: homomorphism indistinguishability over every $\mathsf{CMSO}_1$-definable graph class of bounded cliquewidth is decidable, in randomized exponential time. For classes of bounded linear cliquewidth the bound improves to $\mathsf{PSPACE}$, and we show this is tight by exhibiting such a class for which the problem is $\mathsf{PSPACE}$-complete. These are the first general algorithms for homomorphism indistinguishability over dense graph classes.
