In multi-robot motion planning (MRMP) the aim is to plan the motion of several robots operating in a common workspace, while avoiding collisions with obstacles or with fellow robots. The main contribution of this paper is a simple construction that serves as a lower bound for the computational cost of a variety of prevalent MRMP problems. In particular we show that optimal decoupling of multi-robot motion -- decoupling being a standard approach to practically addressing MRMP -- is NP-hard. The basic problem for which we present our construction is monotone MRMP, a restricted and natural MRMP variant, where robots move one by one to their targets with no intermediate stops. Observing the hardness of these restricted versions of MRMP is significant as it guides the search for efficient solutions towards techniques that can cope with intractable problems. Furthermore, our construction highlights structural properties of difficult instances, such as the need of robots to pass through many start and target positions of other robots. These insights can lead to useful problem relaxations resulting in efficient solutions and to suitable engineering of workspaces.

中文翻译：

---

一类解耦的多机器人运动规划问题的复杂性

在多机器人运动计划（MRMP）中，目标是计划在同一工作空间中操作的多个机器人的运动，同时避免与障碍物或同伴机器人发生碰撞。本文的主要贡献是一种简单的结构，可作为各种流行MRMP问题的计算成本的下限。特别是，我们证明了多机器人运动的最佳去耦-NP是实际解决MRMP的标准方法-很难解决。我们提出的构造的基本问题是单调MRMP，这是一种受限制的自然MRMP变体，机器人在没有中间停止的情况下将机器人逐个移动到其目标。观察这些受限版本的MRMP的硬度非常重要，因为它可以指导人们寻找可解决棘手问题的有效解决方案。此外，我们的结构突出了困难实例的结构特性，例如需要机器人通过其他机器人的许多起始位置和目标位置。这些见解可以导致有用的问题放松，从而产生有效的解决方案，并可以对工作区进行适当的工程设计。