  It is known that the bounded Geodesic Length Problem in free metabelian
groups is NP-complete (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 $Z^2$ 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 $Z^2$.
