Uniform robot relocation is hard in only two directions even without obstacles

Given n unit-sized 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 specific 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. We then consider a relaxed version of this problem called row relocation in which the goal is to move a robot a to a specific row regardless of its horizontal placement. We show that if asking about the first row of the square grid (bottom-most), then this version of the problem is solvable in polynomial time. Finally, we discuss several areas for future research and open problems.