The $k$-dimensional Weisfeiler-Leman ($k$-WL) algorithm is a simple combinatorial algorithm that was originally designed as a graph isomorphism heuristic. It naturally finds applications in Babai's quasipolynomial time isomorphism algorithm, practical isomorphism solvers, and algebraic graph theory. However, it also has surprising connections to other areas such as logic, proof complexity, combinatorial optimization, and machine learning. The algorithm iteratively computes a coloring of the $k$-tuples of vertices of a graph. Since F\"urer's linear lower bound [ICALP 2001], it has been an open question whether there is a super-linear lower bound for the iteration number for $k$-WL on graphs. We answer this question affirmatively, establishing an $\Omega(n^{k/2})$-lower bound for all $k$.

翻译：$k$维Weisfeiler-Leman算法（$k$-WL）是一种简单的组合算法，最初被设计为图同构的启发式方法。它自然地应用于Babai的拟多项式时间同构算法、实用同构求解器以及代数图论中。然而，该算法也与逻辑、证明复杂性、组合优化和机器学习等其他领域存在惊人的联系。该算法迭代计算图中$k$元组顶点的一种着色。自Fürer的线性下界[ICALP 2001]提出以来，关于图上的$k$-WL迭代次数是否存在超线性下界一直是一个开放问题。我们肯定地回答了这一问题，对于所有$k$，建立了$\Omega(n^{k/2})$的下界。