APPROXIMATION OF GEODESICS IN METABELIAN GROUPS

Abstract

It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete [A. Myasnikov, V. Roman'kov, A. Ushakov and A. Vershik, The word and geodesic problems in free solvable groups, Trans. Amer. Math. Soc.362(9) (2010) 4655–4682] (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem. We show that the Geodesic Problem in the restricted wreath product of a finitely generated non-trivial group with a finitely generated abelian group containing ℤ2 is NP-hard and there exists a Polynomial Time Approximation Scheme for this problem. We also show that the Geodesic Problem in the restricted wreath product of two finitely generated non-trivial abelian groups is NP-hard if and only if the second abelian group contains ℤ2.