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

RWTH Publications (RWTH Aachen)

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

2. Moritz Lichter Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University

3. Daniel Neuen

   University of Bremen

   Affiliation as printed

   University of Bremen

4. Pascal Schweitzer

   Technische Universität Darmstadt

   Affiliation as printed

   TU Darmstadt