Efficiently Reconfiguring a Connected Swarm of Labeled Robots

TL;DR

This paper addresses the challenge of reconfiguring a connected swarm of labeled robots efficiently, establishing fundamental bounds and algorithms for optimal reconfiguration times while maintaining connectivity.

Contribution

It proves a lower bound of a ext{(}a ext{(} ext{n}) ext{)} for connected reconfiguration and demonstrates that scaled arrangements allow constant stretch, solving key open problems.

Findings

Established a a ext{(}a ext{(} ext{n}) ext{)} lower bound for connected reconfiguration.

Proved constant stretch is achievable for scaled arrangements of labeled robots.

Decided NP-completeness of achieving a makespan of 2; polynomial check for makespan of 1.

Abstract

When considering motion planning for a swarm of $n$ labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, collision-free robot motions. The objective is to reach the new configuration in a minimum amount of time; an important constraint is to keep the swarm connected at all times. Problems of this type have been considered before, with recent notable results achieving constant stretch for not necessarily connected reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, the total duration of an overall schedule can be bounded to $\mathcal{O}(d)$, which is optimal up to constant factors. However, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations (which arise by increasing all dimensions of a given object by the same multiplicative factor) of unlabeled robots. We resolve these major open problems by (1) establishing a lower bound of $\Omega(\sqrt{n})$ for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected reconfiguration can be achieved. In addition, we show that (3) it is NP-complete to decide whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.