We investigate multiple fundamental variants of the classic coordinated motion planning (CMP) problem for unit square robots in the plane under the $L_1$ metric. In coordinated motion planning, we are given two arrangements of $k$ robots and are tasked with finding a movement schedule that minimizes a certain objective function. The two most prominent objective functions are the sum of distances traveled (Min-Sum) and the latest time of arrival (Min-Makespan). Both objectives have previously been studied extensively. We introduce a new objective function for CMP in the plane. The proposed Min-Exposure objective function defines a set of polygonal regions in the plane that provide cover and asks for a schedule with minimal elapsed time during which at least one robot is partially or fully outside of these regions. We give an $\mathcal{O}(n^4\log n)$ time algorithm that computes exposure-minimal schedules for $k=2$ robots, and an XP algorithm for arbitrary $k$. As a result of independent interest, we leverage new insights to prove that both the Min-Makespan and Min-Sum objectives are fixed-parameter tractable (FPT) parameterized by the number of robots. Our parameterized complexity results generalize known FPT results for rectangular grid graphs [Eiben, Ganian, and Kanj, SoCG'23].

翻译：本文研究了平面上单位正方形机器人在$L_1$度量下经典协调运动规划问题的若干基本变体。在协调运动规划中，给定$k$个机器人的两种初始与目标布局，任务是找到一个运动调度，以最小化特定的目标函数。两个最突出的目标函数是机器人移动距离的总和与最晚到达时间。这两个目标先前均已得到广泛研究。我们为平面上的协调运动规划引入了一种新的目标函数。所提出的最小暴露目标函数定义了平面上提供掩护的一组多边形区域，并要求找到一个调度，使得至少有一个机器人部分或完全处于这些区域之外的总时间最小。我们给出了一个时间复杂度为$\mathcal{O}(n^4\log n)$的算法，用于计算$k=2$个机器人的暴露最小调度，以及一个适用于任意$k$的XP算法。作为一项独立的贡献，我们利用新的见解证明了最小最晚到达时间与最小移动距离和这两个目标函数在机器人数量作为参数时均是固定参数可解的。我们的参数化复杂度结果推广了矩形网格图上已知的固定参数可解性结果[Eiben, Ganian, and Kanj, SoCG'23]。