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 v 1 , v 2 ∈ V , decide whether v 1 and v 2 lie in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories C d (V), d ∈ N = {1, 2, 3,. .. } whose objects include the elements of V. We show that v 1 and v 2 are not in the same orbit by showing that they are not isomorphic in the category C d (V) for some d ∈ 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.