Approximation of Geodesics in Metabelian Groups

TL;DR

This paper studies the computational complexity of the Geodesic Problem in metabelian groups, providing approximation algorithms and complexity results for various group structures.

Contribution

It introduces a 2-approximation polynomial time algorithm for the Geodesic Problem in free metabelian groups and characterizes NP-hardness and approximation schemes in wreath products.

Findings

A 2-approximation polynomial time algorithm for free metabelian groups.

NP-hardness of the Geodesic Problem in certain wreath products.

Existence of a Polynomial Time Approximation Scheme for specific wreath products.

Abstract

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 $Z^2$.