Weisfeiler-Lehman Refinement Requires at Least a Linear Number of Iterations

• Authors: Martin Fürer
• Venue: ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
• Year: 2001

Let Lk, m be the set of formulas of first order logic containing only variables from {x1, x2, ..., xk} and having quantifier depth at most m. Let Ck, m be the extension of Lk, m obtained by allowing counting quantifiers ∃ixj, meaning that there are at least i distinct xj's. It is shown that for constants h ≥ 1, there are pairs of graphs such that h-dimensional Weisfeiler-Lehman refinement (h-dim W-L) can distinguish the two graphs, but requires at least a linear number of iterations. Despite of this slow progress, 2h-dim W-L only requires O(√n) iterations, and 3h-1-dim W-L only requires O(logn) iterations. In terms of logic, this means that there is a c 0 and a class of non-isomorphic pairs (Gnh, Hnh) of graphs with Gnh and Hnh having O(n) vertices such that the same sentences of Lh+1, cn and Ch+1, cn hold (h + 1 variables, depth cn), even though Gnh and Hnh can already be distinguished by a sentence of Lk, m and thus Ck, m for some k h and m = O(logn).