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.

中文翻译：

---

在有限方向上使用统一外部控制重新配置机器人群的硬度

受到纳米级应用和简单机器人代理的推动，当给定有限数量的方向信号和不可移动的几何形状时，我们研究基于使用全局信号移动所有代理的问题。我们研究了一个模型，其中单位正方形粒子基于均匀的外力在 2D 网格内移动。运动基于一系列统一命令，这些命令使所有粒子向特定方向移动 1 步。2D 网格板还包含“阻塞”空间，可防止粒子进入。在这个模型中，我们研究了决定 1) 板上的目标位置是否可以被（任何）粒子占据（\emph{occupancy problem}），2）是否可以将特定粒子重新定位到另一个特定位置的复杂性董事会（\emph{搬迁问题}），和 3) 电路板配置是否可以转换为另一种配置（\emph{reconfiguration problem}）。我们证明，虽然占用率可以在多项式时间内解决，但即使仅限于 2 或 3 个移动方向，重定位和重新配置问题也是 NP-Complete 问题。我们进一步定义了棋盘几何的层次结构，并表明这种硬度甚至适用于非常有限的棋盘几何等级。