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下界。对于$k \geq 3$，我们证明$k$-WL最多经过$O(kn^{k-1}\log n)$次迭代后稳定（其中$n$表示输入结构的顶点数），这首次改进了平凡上界$n^{k}-1$，并扩展了此前针对$k=2$得到的$O(n \log n)$上界[Lichter等人，LICS 2019]。为补充上界结果，我们构造了$k$元关系结构，在此类结构上$k$-WL至少需要$n^{\Omega(k)}$次迭代才能稳定。这改进了此前$n^{\Omega(k / \log k)}$的下界[Berkholz, Nordström, LICS 2016]。我们还研究了WL维度与迭代次数之间的权衡关系，并证明：$d = \lceil\frac{3(k+1)}{2}\rceil$维的$d$-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$d$，其仍需至少$n^{\Omega(k)}$次迭代。$k$-WL区分两个结构所需的迭代次数，对应于一阶逻辑中带计数量词的$(k+1)$变量片段$C_{k+1}$中区分它们的句子的量词秩。因此，我们的结果也蕴含了逻辑$C_{k+1}$中量词秩的新上下界，以及变量数与量词秩之间的权衡关系。