Compressing CFI Graphs and Lower Bounds for the Weisfeiler-Leman Refinements

Journal of the ACM

Abstract

The k -dimensional Weisfeiler-Leman ( k -WL) algorithm is a simple combinatorial algorithm that was originally designed as a graph isomorphism heuristic. It naturally finds applications in Babai’s quasipolynomial-time isomorphism algorithm, practical isomorphism solvers, and algebraic graph theory. However, it also has surprising connections to other areas such as logic, proof complexity, combinatorial optimization, and machine learning. The algorithm iteratively computes a coloring of the k -tuples of vertices of a graph. Since Fürer’s linear lower bound [ICALP 2001], it has been an open question whether there is a super-linear lower bound for the iteration number for k -WL on graphs. We answer this question affirmatively, establishing an Ω ( n k /2 )-lower bound for all k .

Authors 4

1. Martin Grohe Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Aachen, Germany

2. Moritz Lichter Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Aachen, Germany

3. Daniel Neuen

   Max Planck Institute for Informatics

   Affiliation as printed

   Max Planck Institute for Informatics, Saarbrücken, Germany

4. Pascal Schweitzer

   Technische Universität Darmstadt

   Affiliation as printed

   Mathematics, TU Darmstadt, Darmstadt, Germany