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

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. In this talk, I present a novel method of compressing CFI graphs that leads to an \Omega(n^{k/2})-lower bound for all k.

Joint work with Martin Grohe, Moritz Lichter and Pascal Schweitzer (accepted to FOCS 2023).