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

TL;DR

This paper establishes a super-linear lower bound on the number of iterations needed for the $k$-dimensional Weisfeiler-Leman algorithm, advancing understanding of its computational complexity and limitations.

Contribution

It proves an $oldsymbol{ ext{Omega}(n^{k/2})}$ lower bound for the iteration count of $k$-WL, answering an open question in the field.

Findings

Proves a super-linear lower bound for $k$-WL iterations.

Establishes an $oldsymbol{ ext{Omega}(n^{k/2})}$ lower bound for all $k$.

Connects Weisfeiler-Leman bounds to graph isomorphism complexity.

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 F\"urer'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 $\Omega(n^{k/2})$-lower bound for all $k$.