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, Furer 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) is an element of 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, d is an element of 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 for some d is an element of 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-Furer-Immerman graphs in polynomial time. (C) 2013 Elsevier B.V. All rights reserved.