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Computational Complexity of the Weisfeiler-Leman Dimension

Authors Moritz Lichter , Simon Raßmann , Pascal Schweitzer

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Author Details

Moritz Lichter
  • RWTH Aachen University, Germany
Simon Raßmann
  • TU Darmstadt, Germany
Pascal Schweitzer
  • TU 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).
  • Lichter, Moritz: The research of this author has received further funding by the European Union (ERC, SymSim, 101054974).

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Moritz Lichter, Simon Raßmann, and Pascal Schweitzer. Computational Complexity of the Weisfeiler-Leman Dimension. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.13

Abstract

The Weisfeiler-Leman dimension of a graph G is the least number k such that the k-dimensional Weisfeiler-Leman algorithm distinguishes G from every other non-isomorphic graph, or equivalently, the least k such that G is definable in (k+1)-variable first-order logic with counting. The dimension is a standard measure of the descriptive or structural complexity of a graph and recently finds various applications in particular in the context of machine learning. This paper studies the complexity of computing the Weisfeiler-Leman dimension. We observe that deciding whether the Weisfeiler-Leman dimension of G is at most k is NP-hard, even if G is restricted to have 4-bounded color classes. For each fixed k ≥ 2, we give a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph with 5-bounded color classes is at most k. Moreover, we show that for these bounds on the color classes, this is optimal because the problem is PTIME-hard under logspace-uniform AC_0-reductions. Furthermore, for each larger bound c on the color classes and each fixed k ≥ 2, we provide a polynomial-time decision algorithm for the abelian case, that is, for structures of which each color class has an abelian automorphism group.
While the graph classes we consider may seem quite restrictive, graphs with 4-bounded abelian colors include CFI-graphs and multipedes, which form the basis of almost all known hard instances and lower bounds related to the Weisfeiler-Leman algorithm.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity theory and logic
Keywords
  • Weisfeiler-Leman algorithm
  • dimension
  • complexity
  • coherent configurations

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