The Weisfeiler-Lehman (WL) algorithm is an iterative algorithm for certifying that two graphs are not isomorphic. At every iteration a label is generated for each vertex based on the mutliset of labels of its neighbors at the previous iteration. The starting label is the degree of the vertex. When a fixed point is reached the multisets of labels of the vertices of the two graphs are compared. These are different only if the graphs are not isomorphic. In the k-level version of the WL algorithm a label is instead generated for every k-tuple of vertices.
We compare the WL algorithm to a hierarchy of linear programming relaxations of graph isomorphism generated by the Sherali-Adams lift-and-project method. It was already known that two graphs are fractionally isomorphic (i.e. there is a doubly stochastic matrix that converts one graph into the other) if and only if they are not distinguished by the first level of the WL algorithm. We show that, furthermore, the levels of the Sherali-Adams hierarchy interleave in power with the higher levels of the WL algorithm.
Joint work with Albert Atserias.
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