Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithm

Abstract

We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most O(n log n).For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic Weisfeiler-Leman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the walk refinement. This is achieved with an algebraic approach by exploiting properties of semisimple matrix algebras. We also define a walk logic and a bijective walk pebble game that precisely correspond to the new walk refinement.Via the above-mentioned correspondence between the Weisfeiler-Leman (WL) algorithm and the logic [7] we obtain the following corollary.Corollary 2. If two n-vertex graphs can be distinguished by a sentence in 3-variable first order logic with counting C 3 , then there is also a C 3 sentence of quantifier depth at most O(n log n) that distinguishes the two graphs.To prove Theorem 1, we take an algebraic point of view and use a one to one correspondence between coherent configurations and coherent algebras [11]. The WL algorithm produces the former as output, whereas the latter are semisimple matrix algebras closed with respect to the Hadamard multiplication.

The Iteration Number of Colour Refinement

“…Similarly as for Colour Refinement, one can consider the number WL k (n) of iterations of k-WL on graphs of order n. Notably, contrasting our results for Colour Refinement, in [21], it was first proved that the trivial upper bound of WL 2 (n) ≤ n 2 − 1 is not even asymptotically tight (see also the journal version [22]). This foundation fostered further work, leading to an astonishingly good new upper bound of O(n log n) for the iteration number of 2-WL [27].…”

A polynomial time parallel algorithm for graph isomorphism using a quasipolynomial number of processors

“…Consider a graph G = (V, E) with initial coloring C of the vertices. We can expand the initial coloring to 2-tuples by assigning color of C(u, u) by the color of u and C(u, v) based on whether or not (u, v) ∈ E. Then, we employ the idea of walk refinement by Lichter [9]. Given a G = (V, E) with an initial edge coloring C, the k walk refinement is a refinement scheme that for G and C determines a new coloring C W [k] defined by:…”

“…Thus, if u and v are distinguished in the h th iteration of color refinement, then they can be distinguished by a 2h-walk refinement. However, Lichter [9] proved that a 2h-walk refinement can be simulated by log (2h) iterations of 2-dimensional WL refinement.…”

On Jordan schemes

“…This algorithm used nowadays in different modifications is referred to as WL-algorithm or WL-stabilization procedure. The latest results regarding the complexity issues of the WL-algorithm are presented in [13]. In what follows we denote the coherent closure of a set X ⊆ M Ω (F) as W L(X).…”