Bidirectional Sequential Motion Planning

TL;DR

This paper introduces bidirectional topological complexity, a homotopy invariant tailored to motion planning, enabling better estimates of symmetrized topological complexities and offering insights into symmetric motion planning problems.

Contribution

It defines and studies bidirectional topological complexity, a new invariant that simplifies and extends the analysis of symmetric motion planning.

Findings

Properties of bidirectional topological complexity are established.

Instances where symmetrized complexity can be relaxed to the bidirectional setting are identified.

The approach provides estimates for higher symmetrized topological complexities.

Abstract

We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call bidirectional topological complexity. We prove properties of this invariant and show specific instances for which the symmetrized topological complexity can be relaxed to the bidirectional setting. This approach allows us to estimate higher values of symmetrized topological complexities.