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 also consider two modes of motion: serial and parallel. 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. Our techniques for the axis-aligned motion here differ from those for the case of serial motion. We employ a search tree approach and perform a careful examination of the relative geometric positions of the robots that allow us to reduce the problem to FPT-many Linear Programming instances, thus obtaining an FPT algorithm. 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.
