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An Upper Bound on the Weisfeiler-Leman Dimension

Authors Thomas Schneider , Pascal Schweitzer

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LIPIcs.ICALP.2025.129.pdf
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ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Weisfeiler-Leman dimension
  • descriptive complexity
  • coherent configurations

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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 ∈ ℕ there is a corresponding k-dimensional algorithm 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 ∈ ℕ for which 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 ⋅ n + o(n) = 0.15 ⋅ 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.

Cite As Get BibTex

Thomas Schneider and Pascal Schweitzer. An Upper Bound on the Weisfeiler-Leman Dimension. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 129:1-129:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.129

Author Details

Thomas Schneider
  • Department of Mathematics, Technische Universität Darmstadt, Germany
Pascal Schweitzer
  • Department of Mathematics, Technische Universität Darmstadt, Germany

Funding

The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148).

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