Hardness of Reconfiguring Robot Swarms with Uniform External Control in Limited Directions

Motivated by advances is nanoscale applications and simplistic robot agents, we look at problems based on using a global signal to move all agents when given a limited number of directional signals and immovable geometry. We study a model where unit square particles move within a 2D grid based on uniform external forces. Movement is based on a sequence of uniform commands which cause all particles to move 1 step in a specific direction. The 2D grid board additionally contains "blocked" spaces which prevent particles from entry. Within this model, we investigate the complexity of deciding 1) whether a target location on the board can be occupied (by any) particle (\emph{occupancy problem}), 2) whether a specific particle can be relocated to another specific position in the board (\emph{relocation problem}), and 3) whether a board configuration can be transformed into another configuration (\emph{reconfiguration problem}). We prove that while occupancy is solvable in polynomial time, the relocation and reconfiguration problems are both NP-Complete even when restricted to only 2 or 3 movement directions. We further define a hierarchy of board geometries and show that this hardness holds for even very restricted classes of board geometry.