Walk refinement, walk logic, and the iteration number of the WeisfeilerLeman algorithm
Abstract
We show that the 2dimensional WeisfeilerLeman algorithm stabilizes nvertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3variable 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 WeisfeilerLeman 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.03008
 Bibcode:
 2019arXiv190503008L
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics
 EPrint:
 To appear at LICS 19