We prove new upper and lower bounds on the number of iterations the $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) requires until stabilization. For $k \geq 3$, we show that $k$-WL stabilizes after at most $O(kn^{k-1}\log n)$ iterations (where $n$ denotes the number of vertices of the input structures), obtaining the first improvement over the trivial upper bound of $n^{k}-1$ and extending a previous upper bound of $O(n \log n)$ for $k=2$ [Lichter et al., LICS 2019]. We complement our upper bounds by constructing $k$-ary relational structures on which $k$-WL requires at least $n^{\Omega(k)}$ iterations to stabilize. This improves over a previous lower bound of $n^{\Omega(k / \log k)}$ [Berkholz, Nordstr\"{o}m, LICS 2016]. We also investigate tradeoffs between the dimension and the iteration number of WL, and show that $d$-WL, where $d = \lceil\frac{3(k+1)}{2}\rceil$, can simulate the $k$-WL algorithm using only $O(k^2 \cdot n^{\lfloor k/2\rfloor + 1} \log n)$ many iterations, but still requires at least $n^{\Omega(k)}$ iterations for any $d$ (that is sufficiently smaller than $n$). The number of iterations required by $k$-WL to distinguish two structures corresponds to the quantifier rank of a sentence distinguishing them in the $(k + 1)$-variable fragment $C_{k+1}$ of first-order logic with counting quantifiers. Hence, our results also imply new upper and lower bounds on the quantifier rank required in the logic $C_{k+1}$, as well as tradeoffs between variable number and quantifier rank.

翻译：我们证明了 $k$ 维 Weisfeiler-Leman 算法（$k$-WL）在达到稳定状态前所需迭代次数的全新上界与下界。对于 $k \geq 3$，我们证明 $k$-WL 最多在 $O(kn^{k-1}\log n)$ 次迭代后稳定（其中 $n$ 表示输入结构的顶点数），这是对平凡上界 $n^{k}-1$ 的首次改进，并推广了 $k=2$ 时 $O(n \log n)$ 的上界 [Lichter 等人，LICS 2019]。通过构造 $k$ 元关系结构，我们证明在该结构上 $k$-WL 至少需要 $n^{\Omega(k)}$ 次迭代才能稳定，从而改进了先前 $n^{\Omega(k / \log k)}$ 的下界 [Berkholz, Nordström, LICS 2016]。我们还研究了 WL 的维度与迭代次数之间的权衡，并证明当 $d = \lceil\frac{3(k+1)}{2}\rceil$ 时，$d$-WL 仅需 $O(k^2 \cdot n^{\lfloor k/2\rfloor + 1} \log n)$ 次迭代即可模拟 $k$-WL 算法，但对任意（充分小于 $n$ 的）$d$ 而言仍需至少 $n^{\Omega(k)}$ 次迭代。$k$-WL 区分两个结构所需的迭代次数对应于在带计数量词的一阶逻辑 $(k+1)$ 变量片段 $C_{k+1}$ 中区分它们的句子的量词秩。因此，我们的结果也为逻辑 $C_{k+1}$ 所需的量词秩提供了新的上下界，并揭示了变量数与量词秩之间的权衡关系。