An Upper Bound on the Weisfeiler-Leman Dimension

TL;DR

This work delivers an explicit upper bound on the WL-dimension of graphs with $n$ vertices, showing $\mathrm{WLdim}(G) \le \frac{3}{20}n + o(n)$ by lifting the problem to coherent configurations and applying a disciplined hierarchy of local and global reductions. Central to the method is the introduction of a potential function $\tau$ that tracks progress, the notion of critical and restorable fibers, and a finite classification of interspaces between small fibers, all enabling a recursive reduction to base cases where fibers are tiny or of bounded size. The authors couple local identifications with global decompositions, leveraging treewidth bounds on quotient graphs and Zemlyachenko-style degree reductions to bound WL-dimension in substructures, then aggregate these bounds to reach the global result. They also establish a complementary lower bound, showing that WL-dimension can be as large as $0.0105027\cdot n - o(n)$, confirming nontrivial growth and the tightness of the approach up to constants. Overall, the paper advances descriptive complexity bounds for graph isomorphism, with implications for fixed-point counting logics and algorithmic graph analysis, and introduces techniques potentially adaptable to broader WL-like frameworks.

Abstract

The Weisfeiler-Leman (WL) algorithms form a family of incomplete approaches to the graph isomorphism problem. They recently found various applications in algorithmic group theory and machine learning. In fact, the algorithms form a parameterized family: for each $k \in \mathbb{N}$ there is a corresponding $k$-dimensional algorithm $\texttt{WLk}$. The algorithms become increasingly powerful with increasing dimension, but at the same time the running time increases. The WL-dimension of a graph $G$ is the smallest $k \in \mathbb{N}$ for which $\texttt{WLk}$ correctly decides isomorphism between $G$ and every other graph. In some sense, the WL-dimension measures how difficult it is to test isomorphism of one graph to others using a fairly general class of combinatorial algorithms. Nowadays, it is a standard measure in descriptive complexity theory for the structural complexity of a graph. We prove that the WL-dimension of a graph on $n$ vertices is at most $3/20 \cdot n + o(n) = 0.15 \cdot n + o(n)$. Reducing the question to coherent configurations, the proof develops various techniques to analyze their structure. This includes sufficient conditions under which a fiber can be restored uniquely up to isomorphism if it is removed, a recursive proof exploiting a degree reduction and treewidth bounds, as well as an exhaustive analysis of interspaces involving small fibers. As a base case, we also analyze the dimension of coherent configurations with small fiber size and thereby graphs with small color class size.

Figures (5)

• Figure 1: The graph ${\overrightarrow{C_3}[K_2]}$.
• Figure 2: The graph $\mathop{\mathrm{L}}\nolimits\!\left(\mathop{\mathrm{FP}}\nolimits\right)$.
• Figure 3: The graph $Sp_{4,6}$.
• Figure 4: Coherent configurations of order $6$ with directed edges.
• Figure 5: Visualisation of the partition of the neighborhood in Lemma \ref{['critical:6-cc:restorable:large-neighborhood/lem']}.

Theorems & Definitions (126)

• Theorem 2.1
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof : Proof sketch
• Lemma 4.4
• proof : Proof sketch
• Lemma 4.5
• proof
• Lemma 4.6