Towards faster isomorphism tests for bounded-degree graphs
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Luks' algorithm (JCSS 1982) to test isomorphism of bounded degree graphs in polynomial time is one of most important results in the context of the Graph Isomorphism Problem and has been repeatedly used as a basic building block for many other algorithms. In the light of Babai's (STOC 2016) quasipolynomial time algorithm for general isomorphism testing we investigate the question whether the techniques developed for this new algorithm can adapted to the bounded degree setting. More precisely, we ask the question whether there is an n^polylog(d)-isomorphism test for graphs of maximum degree d. While we cannot answer this questions yet, we prove that there is an n^polylog(d)-isomorphism test for all classes of graphs of maximum degree d and diameter at most O( n). In particular, this implies that there is an n^polylog(d)-isomorphism test for all families of d-regular expander graphs. We derive this result from a more general (and quite technical) theorem about the so-called string isomorphism problem for a family of groups that also arises in Luks' work. This theorem has also other interesting applications, for example to the isomorphism problem for relational structures.
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