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：本文研究计算上困难的基础运动规划问题，目标是在无碰撞条件下将$k$个轴对齐矩形机器人从初始位置平移至最终位置，并最小化平移移动次数。我们的目标是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位数）之间的关联。重点考虑轴对齐平移，以及更一般地限制在给定方向集合内的平移情形，并研究两种场景：机器人在自由平面中移动以及被限制在边界框内移动。针对所有考虑的场景，我们获得了以$k$为参数的固定参数可解（FPT）算法。在机器人串行移动（即每个时间步仅移动一个机器人）且为轴对齐平移的情况下，我们证明了一个结构性结论：每个问题实例均存在一个最优解，其中移动沿网格进行，该网格的规模是$k$的函数，且可基于输入实例定义。这一结构性结论表明该问题在参数$k$下是固定参数可解的。我们还考虑了机器人并行移动（即同一时间步内多个机器人可同时移动）的情形，该情形属于协调运动规划问题范畴。最后，我们证明：当机器人在自由平面中移动时，串行移动情形的FPT结果可推广至平移方向限制在任意给定方向集合中的情形。