Some geodesic problems in groups

TL;DR

This paper investigates the computational complexity of various geodesic problems in finitely generated groups, demonstrating polynomial-time reductions among known problems and introducing new related problems.

Contribution

It establishes polynomial-time equivalences among three existing geodesic problems and explores two novel geodesic problems in group theory.

Findings

The three geodesic problems are polynomial-time reducible to each other.

New geodesic problems are introduced and analyzed.

Provides insights into the computational complexity of geodesic problems in groups.

Abstract

We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two new geodesic problems which arise in a previous paper of the authors and Fusy [arXiv:0902.0202] .