Uniform Robot Relocation Is Hard in only Two Directions Even Without Obstacles.

Given n robots contained within a square grid surrounded by four walls, we ask the question of whether it is possible to move a particular robot a to a particular grid location b by performing a sequence of global step operations in which all robots move one grid step in the same cardinal direction (if not blocked by a wall or other blocked robots). We show this problem is NP-complete when restricted to just two directions (south and west). This answers the simplest fundamental problem in uniform global unit tilt swarm robotics.