The Graph Isomorphism Problem and approximate categories

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler–Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Fürer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G   and two elements v1,v2∈Vv1,v2∈V, decide whether v1v1 and v2v2 lie in the same G  -orbit. Then we attack the Orbit Problem by constructing certain approximate categories CdCd, d∈N={1,2,3,…}d∈N={1,2,3,…} whose objects include the elements of V  . We show that v1v1 and v2v2 are not in the same orbit by showing that they are not isomorphic in the category CdCd for some d∈Nd∈N. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler–Lehman algorithm, our algorithm can distinguish the Cai–Fürer–Immerman graphs in polynomial time.