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arXiv 2402.11531cs.CCcs.DM
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Weisfeiler-Leman维度的计算复杂度

Computational complexity of the Weisfeiler-Leman dimension

Moritz Lichter, Simon Raßmann, Pascal Schweitzer

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AI总结：

本文研究了计算图Weisfeiler-Leman维度的复杂度，证明了其判定问题为NP-hard，并针对颜色重数受限的图类提出了最优的多项式时间判定算法。

摘要：

图$G$的Weisfeiler-Leman维度是指使得$k$维Weisfeiler-Leman算法能将$G$与其他所有非同构图区分开来的最小数字$k$。该维度是图描述复杂度的标准度量，近期在机器学习领域找到了各种应用。在本文中，我们研究了计算Weisfeiler-Leman维度的计算复杂度。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-hard的。对于颜色重数至多为4的图的更受限问题，这一点同样成立。因此，我们研究了该问题的参数化版本。对于每个固定的$k\geq 2$，我们给出了一种多项式时间算法，用于判定颜色重数至多为5的给定图的Weisfeiler-Leman维度是否至多为$k$。此外，我们证明了对于这些颜色重数，这在logspace-uniform $\text{AC}_0$-reductions下是P-hard的意义上是最优的。进一步地，对于颜色类的每个更大边界$c$以及每个固定的$k\geq 2$，我们为阿贝尔情况提供了多项式时间判定算法，即对于每个颜色类具有阿贝尔自同构群的结构。虽然我们考虑的图类可能看起来相当受限，但具有4-bounded abelian colors的图包含了CFI-graphs和multipedes，它们构成了几乎所有已知的与Weisfeiler-Leman算法相关的困难实例和下界的基础。

英文摘要：

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.

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