2001 | OriginalPaper | Buchkapitel
Weisfeiler-Lehman Refinement Requires at Least a Linear Number of Iterations
verfasst von : Martin Fürer
Erschienen in: Automata, Languages and Programming
Verlag: Springer Berlin Heidelberg
Enthalten in: Professional Book Archive
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Let L k , m be the set of formulas of first order logic containing only variables from x1, x2, . . . , x K and having quantifier depth at most m. Let Ck,m be the extension of L k,m obtained by allowing counting quantifiers ∃ix j , meaning that there are at least i distinct x j ’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(log n) iterations. In terms of logic, this means that there is a c > 0 AND A CLASS OF NON-ISOMORPHIC PAIRS $$ (\mathop G\nolimits_{n,}^h \mathop H\nolimits_n^h ) $$ of graphs with $$ \mathop G\nolimits_n^h and \mathop H\nolimits_n^h $$ 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 $$ \mathop G\nolimits_n^h and \mathop H\nolimits_n^h $$ can already be distinguished by a sentence of L k,m and thus C k,m for some k > h and m = O(log n).