The Iteration Number of the Weisfeiler-Leman Algorithm

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 ≥ 3, we show that k-WL stabilizes after at most O(knk−1log n) iterations (where n denotes the number of vertices of the input structures), obtaining the first improvement over the trivial upper bound of nk− 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 nΩ(k)iterations to stabilize. This improves over a previous lower bound of nΩ(k/logk)[Berkholz, Nordström, LICS 2016].We also investigate tradeoffs between the dimension and the iteration number of WL, and show that d-WL, where $d = \left\lceil {\frac{{3(k + 1)}}{2}} \right\rceil$, can simulate the k-WL algorithm using only O(k2• n⌊k/2⌋+1log n) many iterations, but still requires at least nΩ(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 ${{\mathcal{C}}_k}_{ + 1}$ 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 ${{\mathcal{C}}_k}_{ + 1}$, as well as tradeoffs between variable number and quantifier rank.