The Iteration Number of the Weisfeiler-Leman Algorithm

TL;DR

This work analyzes the iteration number of the k-dimensional Weisfeiler-Leman algorithm, establishing a nontrivial upper bound of $O(k n^{k-1}\log n)$ rounds for all $k\ge 2$ and constructing $k$-ary structures that require at least $n^{\Omega(k)}$ rounds, improving prior lower bounds. It also studies the tradeoff between the WL dimension and iteration depth, showing that increasing dimension to $d=\lceil 3(k+1)/2\rceil$ can simulate $k$-WL with $O(k^2 n^{\lfloor k/2\rfloor+1}\log n)$ rounds, while still needing $n^{\Omega(k)}$ rounds in general. The results translate into bounds on the quantifier rank in the $(k+1)$-variable fragment of counting logic and reveal intrinsic limits on low-dimension logics to distinguish certain structures. Overall, the paper advances our understanding of the expressive power and limits of WL-based methods in descriptive complexity and has implications for graph isomorphism heuristics and logic-based characterizations of polynomial-time properties.

Abstract

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 $n^{Ω(k)}$ iterations to stabilize. This improves over a previous lower bound of $n^{Ω(k / \log k)}$ [Berkholz, Nordström, 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 $n^{Ω(k)}$ 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.

Figures (1)

• Figure 1: Visualization of $(\ell \times m)$-layered graphs.

Theorems & Definitions (77)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Corollary 2.2
• Theorem 2.3: LichterPS19
• proof