The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive complexity of a graph and recently finds various applications in particular in the context of machine learning. In this paper, we study the computational complexity of computing the Weisfeiler-Leman dimension. We observe that in general the problem of deciding whether the Weisfeiler-Leman dimension of $G$ is at most $k$ is NP-hard. This is also true for the more restricted problem with graphs of color multiplicity at most 4. Therefore, we study parameterized versions of the problem. We give, for each fixed $k\geq 2$, a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph of color multiplicity at most $5$ is at most $k$. Moreover, we show that for these color multiplicities this is optimal in the sense that this problem is P-hard under logspace-uniform $\text{AC}_0$-reductions. Furthermore, for each larger bound $c$ on the color classes and each fixed $k\geq 2$, we provide a polynomial-time decision algorithm for the abelian case, that is, for structures of which each color class has an abelian automorphism group. While the graph classes we consider may seem quite restrictive, graphs with $4$-bounded abelian colors include CFI-graphs and multipedes, which form the basis of almost all known hard instances and lower bounds related to the Weisfeiler-Leman algorithm.

翻译：图$G$的Weisfeiler-Leman维度是指最小的数$k$，使得$k$维Weisfeiler-Leman算法能将$G$与所有其他非同构图区分开来。该维度是衡量图描述复杂性的标准指标，近年来尤其在机器学习领域得到了广泛应用。本文研究计算Weisfeiler-Leman维度的计算复杂性。我们观察到，一般而言，判定图$G$的Weisfeiler-Leman维度是否至多为$k$的问题是NP困难的。对于颜色重数至多为4的图，该问题同样成立。因此，我们研究该问题的参数化版本。对于每个固定的$k\geq 2$，我们给出了一个多项式时间算法，用于判定给定颜色重数至多为$5$的图的Weisfeiler-Leman维度是否至多为$k$。此外，我们证明对于这些颜色重数，该结果是最优的，即该问题在对数空间均匀$\text{AC}_0$归约下是P困难的。进一步地，对于颜色类的每个更大上界$c$以及每个固定的$k\geq 2$，我们为阿贝尔情形（即每个颜色类都具有阿贝尔自同构群的结构）提供了一个多项式时间判定算法。尽管我们考虑的图类可能看似限制性较强，但具有4-有界阿贝尔颜色的图包括CFI图和蜈蚣图，它们构成了几乎所有已知与Weisfeiler-Leman算法相关的困难实例和下界的基础。