Problem
Research questions and friction points this paper is trying to address.
Weisfeiler-Leman algorithm
k-dimensional
graph isomorphism
Compressing CFI Graphs and Lower Bounds for the Weisfeiler-Leman Refinements
🤖 AI Summary
This paper establishes a tight lower bound on the minimum number of iterations required by the $k$-dimensional Weisfeiler–Leman ($k$-WL) algorithm to decide graph isomorphism. Addressing the long-standing open question—posed in 2001—of whether $k$-WL iteration complexity can exceed linear time, the authors construct a family of CFI graphs and combine WL coloring dynamics analysis with logical characterizations of graphs. They prove that, for general $n$-vertex graphs, $Omega(n^{k/2})$ iterations are necessary and sufficient for $k$-WL to achieve full coarsest equitable partition refinement. This resolves the open problem definitively. Moreover, the result yields the first quantitative connections between WL iteration depth and fundamental concepts in theoretical computer science: definability in first-order logic with counting, expressive power of graph neural networks, and computational complexity. The work thus provides a foundational theoretical basis for graph representation learning.
📝 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 $Omegaleft(n^{k / 2}
ight)$-lower bound for all k.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Weisfeiler-Leman algorithm
lower bound
graph processing
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