Bren Cavallo

A family of polycyclic groups over which the uniform conjugacy problem is NP-complete

In this paper, we study the conjugacy problem in polycyclic groups. Our main result is that we construct polycyclic groups Gn whose conjugacy problem is at least as hard as the subset sum problem with n indeterminates. As such, the uniform conjugacy problem over the groups Gn is NP-complete where the parameters of the problem are taken in terms of n and the length of the elements given on input.

Erratum: A family of polycyclic groups over which the uniform conjugacy problem is NP-complete

In [1] we attempted to construct a family of polycyclic groups, {Gn}, such that the uniform conjugacy problem over the {Gn} NP-Complete . Unfortunately, it was brought to our attention by the Board that our construction detailed in Section 6 of [1] is flawed in that the maps we use to extend Gi to obtain Gi+1 are not automorphisms which in turn implies that the twisted subset problem (described in Section 7 of the same paper) we obtain following conjugation is trivial. For instance, when we extend G1 to G2 we have the following problem which we describe in the notation given in Section 6 of [1]. Consider the following words given by the generators of G2: g4g2(g1g3)g−1 2 g −1 4 = g2g4(g1g3)g −1 4 g −1 2