We consider the problem of connected coordinated motion planning for a large collective of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-complete, even for deciding 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. On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, then the total duration of our overall schedule is $\mathcal{O}(d)$, which is optimal up to constant factors.

翻译：我们研究大量简单、相同机器人的连通协调运动规划问题：从给定的机器人初始网格配置出发，需要通过一系列并行的、无碰撞的机器人运动达到期望的目标配置，且在所有整数时刻机器人的集合均诱导出连通的网格图。目标是最小化运动调度的时间跨度，即尽可能快地到达新配置。我们证明该问题是NP完全的，即使判断时间跨度是否为2也是NP完全的，而判断时间跨度是否为1则可在多项式时间内完成。在算法方面，我们通过实现常数的伸缩和尺度，建立了针对两个基本参数的同步常数因子近似。伸缩形状（通过将给定物体的所有维度乘以相同倍数得到）已出现在自组装领域的开创性工作中，通常使用无界或对数尺度因子；我们为有界于常数的广义尺度因子提供了方法。此外，我们的算法实现了常数伸缩因子：若将初始配置映射到目标配置所需的最大曼哈顿距离为$d$，则整体调度的总持续时间为$\mathcal{O}(d)$，这在常数因子意义下是最优的。