ScrewBox: a Randomized Certifying Graph-Non-Isomorphism Algorithm

Abstract

We present a novel randomized approach to the graph isomorphism problem. Our algorithm aims at solving difficult instances by producing randomized certificates for non-isomorphism. We compare our implementation to the de facto standard nauty. On many of the hardest known instances, the incidence graphs of finite projective planes, our program is considerably faster than nauty. However, it is inherent to our approach that it performs better on pairs of non-isomorphic graphs than on isomorphic instances. Our algorithm randomly samples substructures in the given graphs in order to detect dissimilarities between them. The choice of the sought-after structures as well as the tuning of the search process is dynamically adapted during the sampling. Eventually, a randomized certificate is produced by which the user can verify the non-isomorphism of the input graphs. As a byproduct of our approach, we introduce a new concept of regularity for graphs which is meant to capture the computational hardness of isomorphism problems on graphs.