Some elements of a Galois theory of the structure and complexity of the tree automorphism problem

Abstract

Engeler's generalization of Galois theory is applied to the tree automorphism problem. We compute the Galois group of each instance of this problem. The group yields information on four aspects of problem difficulty: lower bounds for the time complexity of different solution approaches, ‘hard’ instances of the problem, dependence of problem difficulty on structural parameters of the input and relative solvability. In addition we show that an approximation of this information can be obtained from the group of the problem of finding ‘approximate’ tree automorphisms.