Let $\mathcal{W} \subset \mathbb{R}^2$ be a rectilinear polygonal environment (that is, a rectilinear polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can move rectilinearly inside $\mathcal{W}$. The goal is to compute a collision-free motion plan $\boldsymbolπ$, that is, a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. We study two variants of this problem which are focused additionally on the optimality of $\boldsymbolπ$, and obtain the following results. 1. Min-Sum: Here the goal is to compute a motion plan that minimizes the sum of the lengths of the paths of the robots. We present an $O(n^4\log{n})$-time algorithm for computing an optimal solution to the min-sum problem. This is the first polynomial-time algorithm to compute an optimal, collision-free motion of two robots amid obstacles in a planar polygonal environment. 2. Min-Makespan: Here the robots can move with at most unit speed, and the goal is to compute a motion plan that minimizes the maximum time taken by a robot to reach its target location. We prove that the min-makespan variant is NP-hard.

翻译：设 $\mathcal{W} \subset \mathbb{R}^2$ 为一个矩形多边形环境（即一个可能包含孔洞的矩形多边形），其顶点总数为 $n$，并设 $A$、$B$ 为两个机器人，每个均建模为一个轴对齐的单位正方形，可在 $\mathcal{W}$ 内进行直线运动。目标是计算一个无碰撞运动规划 $\boldsymbolπ$，即一个连续地将 $A$ 从 $s_A$ 移动到 $t_A$、将 $B$ 从 $s_B$ 移动到 $t_B$ 的运动规划，使得 $A$ 和 $B$ 在整个运动过程中始终保持在 $\mathcal{W}$ 内部且彼此不发生碰撞。我们研究了该问题的两个变体，这些变体额外关注 $\boldsymbolπ$ 的最优性，并获得了以下结果。1. 最小和：此处的目标是计算一个运动规划，使得机器人路径长度之和最小。我们提出了一个时间复杂度为 $O(n^4\log{n})$ 的算法，用于计算最小和问题的最优解。这是首个在平面多边形障碍环境中计算两个机器人最优无碰撞运动的多项式时间算法。2. 最小完工时间：此处机器人可以以不超过单位速度移动，目标是计算一个运动规划，使得机器人到达其目标位置所需的最大时间最小。我们证明了最小完工时间变体是 NP 难的。