A General backtrack algorithm for the isomorphism problem of combinatorial objects

Abstract

Our aim is to present a practical algorithm for the isomorphism problem that can be easily adapted to any class of combinatorial objects. We investigate the underlying principles of backtrack algorithms that determine a canonical representative of a combinatorial object. We identify the parts of the algorithm that are dependent on the class of combinatorial objects and those parts that are independent of the class. An interface between the two parts is developed to provide a general backtrack algorithm for the isomorphism problem of combinatorial objects that incorporates the technique of branch-and-bound, and that also uses the automorphisms of the combinatorial object to prune the search tree. Our general algorithm incorporates from computational group theory an algorithm known as the base change algorithm. The base change algorithm allows one to recover as much information as possible about the automorphism group when a new branch of the search tree is processed. Thus, it can lead to greater pruning of the search tree. This work is intended to lead to a better understanding of the practical isomorphism algorithms. It is not intended as a contribution to the theoretical study of the complexity of the isomorphism problem.