Topological complexity and efficiency of motion planning algorithms

TL;DR

This paper introduces a new variant of topological complexity for Riemannian manifolds that measures the efficiency of motion planners based on average path length, showing it closely relates to the classical invariant.

Contribution

It defines an efficient topological complexity for smooth manifolds and proves its close relationship to the classical topological complexity invariant.

Findings

The new invariant differs from classical topological complexity by at most 1.

Efficient motion planners exist with average path length close to the minimal possible.

The invariant provides a measure of the existence of efficient motion planning algorithms.

Abstract

We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove that it never differs from topological complexity by more than $1$, thus showing that the latter invariant addresses the problem of the existence of motion planners which are "efficient".