eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2012-09-03
289
304
10.4230/LIPIcs.CSL.2012.289
article
Pebble Games and Linear Equations
Grohe, Martin
Otto, Martin
We give a new, simplified and detailed account of the correspondence
between levels of the Sherali-Adams relaxation of graph isomorphism
and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Ramana, Scheinerman and Ullman, is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva, who prove that the two resulting hierarchies interleave.
In carrying this analysis further, we here give (a) a precise characterisation of the level-k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict.
We also investigate the variation based on boolean arithmetic instead
of real/rational arithmetic and obtain analogous correspondences and
separations for plain k-pebble equivalence (without counting). Our
results are driven by considerably simplified accounts of the
underlying combinatorics and linear algebra.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol016-csl2012/LIPIcs.CSL.2012.289/LIPIcs.CSL.2012.289.pdf
Finite model theory
finite variable logics
graph isomorphism
Weisfeiler- Lehman algorithm
linear programming
Sherali–Adams hierarchy