In the coordinated motion planning problem, we are given a graph together with the starting and destination vertices of $k$ robots. At each time step, any subset of robots may move, each traversing an edge of the graph, provided that no two robots collide. The goal is to compute a schedule that routes all robots to their destinations while minimizing some objective function. In this paper, we focus on the well-studied objective of minimizing the total travel length of all robots. This problem is known to be NP-hard, and it has been shown to be fixed-parameter tractable (FPT), when parameterized by the number $k$ of robots, on full grids (SoCG 2023) and on bounded-treewidth graphs (ICALP 2024). We present a fixed-parameter algorithm for coordinated motion planning, parameterized by the number $k$ of robots, on graphs arising from discretizations of simple polygons. Such graphs are of particular interest in real-world applications, where planar motion is often constrained to discretized representations of polygonal environments. Moreover, these graphs generalize rectangular grids; consequently, our result constitutes a significant step toward resolving the parameterized complexity of coordinated motion planning on subgrids and, ultimately, planar graphs -- two prominent open problems in the field.

翻译：在协调运动规划问题中，给定一个图以及 $k$ 个机器人的起点和终点顶点。在每个时间步，任意子集的机器人可以移动，每个机器人沿图的一条边移动，前提是任意两个机器人不发生碰撞。目标是计算一个将所有机器人调度至其目的地的方案，同时最小化某个目标函数。本文关注一个被广泛研究的目标：最小化所有机器人的总行进距离。该问题已知为NP困难，且在完全网格（SoCG 2023）和有界树宽图（ICALP 2024）上已被证明在以机器人数量 $k$ 为参数时是固定参数可处理的（FPT）。我们针对源自简单多边形离散化的图上的协调运动规划问题，提出了一个以机器人数量 $k$ 为参数的固定参数算法。这类图在实际应用中尤为重要，因为平面运动常受限于多边形环境的离散化表示。此外，这些图推广了矩形网格；因此，我们的结果标志着向着解决子网格图及最终平面图上的协调运动规划的参数复杂性这两个领域的突出未解决问题迈出了重要一步。