On the Complexity of a Family of Decoupled Multi-Robot Motion Planning Problems

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.