fractional isomorphism
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2015 ◽  
Vol 80 (3) ◽  
pp. 797-844 ◽  
Author(s):  
MARTIN GROHE ◽  
MARTIN OTTO

AbstractWe 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 Tinhofer [22; 23] and Ramana, Scheinerman and Ullman [17], is re-interpreted as the base level of Sherali–Adams and generalised to higher levels in this sense by Atserias and Maneva [1] and Malkin [14], 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.



2007 ◽  
Vol 56 (1) ◽  
pp. 17-32
Author(s):  
A. D. Forbes ◽  
M. J. Grannell ◽  
T. S. Griggs


1994 ◽  
Vol 132 (1-3) ◽  
pp. 247-265 ◽  
Author(s):  
Motakuri V. Ramana ◽  
Edward R. Scheinerman ◽  
Daniel Ullman


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