2026-07-21T20:03:26Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23506 2025-10-02T12:57:11Z ddc:004 open_access

An Upper Bound on the Weisfeiler-Leman Dimension Schneider, Thomas Schweitzer, Pascal Weisfeiler-Leman dimension descriptive complexity coherent configurations 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Thomas Schneider and Pascal Schweitzer 2025 Is Part Of LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2025.129 urn:nbn:de:0030-drops-235065 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.129 eng https://creativecommons.org/licenses/by/4.0/legalcode