We study computationally-hard fundamental motion planning problems where the goal is to translate $k$ axis-aligned rectangular robots from their initial positions to their final positions without collision, and with the minimum number of translation moves. Our aim is to understand the interplay between the number of robots and the geometric complexity of the input instance measured by the input size, which is the number of bits needed to encode the coordinates of the rectangles' vertices. We focus on axis-aligned translations, and more generally, translations restricted to a given set of directions, and we study the two settings where the robots move in the free plane, and where they are confined to a bounding box. We obtain fixed-parameter tractable (FPT) algorithms parameterized by $k$ for all the settings under consideration. In the case where the robots move serially (i.e., one in each time step) and axis-aligned, we prove a structural result stating that every problem instance admits an optimal solution in which the moves are along a grid, whose size is a function of $k$, that can be defined based on the input instance. This structural result implies that the problem is fixed-parameter tractable parameterized by $k$. We also consider the case in which the robots move in parallel (i.e., multiple robots can move during the same time step), and which falls under the category of Coordinated Motion Planning problems. Finally, we show that, when the robots move in the free plane, the FPT results for the serial motion case carry over to the case where the translations are restricted to any given set of directions.

翻译：我们研究计算上困难的基本运动规划问题，目标是将$k$个轴对齐矩形机器人从初始位置平移至最终位置，且无碰撞并使平移步数最小。旨在理解机器人数量与输入实例几何复杂度（即编码矩形顶点坐标所需比特数表示的输入规模）之间的相互作用。重点关注轴对齐平移，以及更一般地限制在给定方向集合上的平移，并研究机器人在自由平面运动及被限制于边界框内运动两种场景。针对所有考虑场景，我们获得了以$k$为参数的固定参数易解（FPT）算法。在机器人串行移动（即每一步仅一个机器人移动）且轴对齐的情况下，我们证明了一个结构性结论：每个问题实例均存在最优解，其中移动沿网格进行，该网格的大小是$k$的函数且可基于输入实例定义。该结构性结论表明问题关于参数$k$是固定参数易解的。同时考虑机器人并行移动（即同一时间步允许多个机器人移动）的场景，该场景属于协调运动规划问题范畴。最后证明，当机器人在自由平面运动时，串行运动场景的FPT结果可推广至平移受限为任意给定方向集合的情形。