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

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

arXiv comment26 pages, 5 figures, full version of a paper accepted at FOCS 2023 v2: fixed mistake in the definition of a compression (Definition 7)

References in corpus (5)

• Separating Rank Logic from Polynomial Time
• Oddomorphisms and homomorphism indistinguishability over graphs of bounded degree
• The Iteration Number of the Weisfeiler-Leman Algorithm
• Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Graph Isomorphism Problem
• Truly Supercritical Trade-offs for Resolution, Cutting Planes, Monotone Circuits, and Weisfeiler-Leman