Upper Bounds on the Quantifier Depth for Graph Differentiation in First-Order Logic

TL;DR

This paper establishes tighter upper bounds on the number of iterations needed for the Weisfeiler-Leman algorithm to stabilize on graphs, and translates these bounds into quantifier depth limits for distinguishing graphs in first-order logic with counting.

Contribution

It provides the first asymptotically tight upper bound on Weisfeiler-Leman iterations and quantifier depth for graph differentiation in first-order logic with counting.

Findings

Weisfeiler-Leman stabilizes in O(n^2/log(n)) iterations on graphs with n vertices.

Graphs with bounded color class size stabilize in linear time.

Distinguishing graphs in C3 logic requires quantifier depth at most O(n^2/log(n)).

Abstract

We show that on graphs with n vertices, the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n^2/log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n^2) is asymptotically not tight. In the logic setting, this translates to the statement that if two graphs of size n can be distinguished by a formula in first-order logic with counting with 3 variables (i.e., in C3), then they can also be distinguished by a C3-formula that has quantifier depth at most O(n^2/log(n)). To prove the result we define a game between two players that enables us to decouple the causal dependencies between the processes happening simultaneously over several iterations of the algorithm. This allows us to treat large color classes and small color classes separately. As part of our proof we show that for graphs with bounded color class size, the number of iterations until stabilization is at most linear in the number of vertices. This also yields a corresponding statement in first-order logic with counting. Similar results can be obtained for the respective logic without counting quantifiers, i.e., for the logic L3.