经典 Weisfeiler-Leman 算法在 $O(n)$ 轮内稳定
The Classical Weisfeiler-Leman Algorithm Stabilizes in $O(n)$ Rounds
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中文总结 AI 辅助
该论文证明经典 Weisfeiler-Leman 算法在 $5(n-1)$ 轮内终止，改进上界至 $O(n)$，并推广到 $k$ 维情形，获得更紧的迭代次数上界。
中文摘要 AI 辅助
经典 Weisfeiler-Leman 算法（也称为 $2$ 维 Weisfeiler-Leman 算法）是一种简单的组合算法，最初被设计为图同构问题的启发式方法。然而，它还与代数图论、逻辑学、证明复杂性、组合优化和机器学习等其他领域有着众多联系。我们证明了经典 Weisfeiler-Leman 算法在 $5(n-1)$ 次迭代后终止。这改进了 Lichter、Ponomarenko 和 Schweitzer [LICS 2019] 先前的最佳上界 $O(n \log n)$，并在渐近意义上匹配 Fürer [ICALP 2001] 已知的下界 $\Omega(n)$。此外，基于我们对 $2$ 维情形的结果，对于每个 $k \geq 3$，我们获得了 $k$ 维 Weisfeiler-Leman 算法迭代次数的改进上界 $O(n^{k-1}/(k-2)! + n^{k-2})$。我们的论证实际上适用于更广泛的 $k$ 元组着色序列类别；在这个更广泛的类别中，我们的上界对于所有 $k \geq 3$ 本质上是紧的。
英文摘要
The classical Weisfeiler-Leman algorithm (also known as the $2$-dimensional Weisfeiler-Leman algorithm) is a simple combinatorial algorithm that was originally designed as a heuristic for the graph isomorphism problem. However, it has also numerous connections to other areas such as algebraic graph theory, logics, proof complexity, combinatorial optimization and machine learning. We prove that the classical Weisfeiler-Leman algorithm terminates after $5(n-1)$ iterations. This improves over the previous best upper bound of $O(n \log n)$ by Lichter, Ponomarenko and Schweitzer [LICS 2019], and asymptotically matches the known lower bound of $Ω(n)$ by Fürer [ICALP 2001]. Additionally, building on our results for the $2$-dimensional case, we obtain an improved upper bound of $O(n^{k-1}/(k-2)! + n^{k-2})$ on the number of iterations performed by the $k$-dimensional Weisfeiler-Leman algorithm, for every $k \geq 3$. Our arguments actually hold for a larger class of sequences of colorings of $k$-tuples; in this larger class our upper bounds are essentially tight for all $k \geq 3$.
发表机构
- Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
- Saarland University (SIC)（萨尔兰大学）
- TU Dresden（德累斯顿工业大学）
机构由 AI 辅助整理，请以论文原文为准。
