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.

翻译：在考虑$n$个带标签机器人的集群运动规划时，我们需要通过一系列并行、无碰撞的机器人运动，将给定的初始构型重排为目标构型。目标是在最短时间内达到新构型；一个重要约束是始终保持集群的连通性。此类问题先前已被研究，近期显著成果在非连通重构场景下实现了常数拉伸：若将初始构型映射至目标构型所需的最大曼哈顿距离为$d$，则整体调度总时长可被限定在$\mathcal{O}(d)$内，该结果在常数因子范围内是最优的。然而，常数拉伸仅能在允许非连通重构的情况下实现，或针对未标签机器人的缩放构型（通过对给定对象所有维度施加相同乘性因子放大而产生）。我们通过以下工作解决了这些重要开放问题：(1) 证明连通带标签重构的下界为$\Omega(\sqrt{n})$；最关键的是，(2) 证明对于缩放构型，连通重构可以实现常数拉伸。此外，我们表明：(3) 判定能否实现2的总完工时间是NP完全问题，而能否实现1的总完工时间可在多项式时间内验证。