Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment (i.e., a 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 translate inside $\mathcal{W}$. Given source and target placements $s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to compute a \emph{collision-free motion plan} $\mathbf{\pi}^*$, i.e., 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. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots, $\left|\mathbf{\pi}^*\right|$. Given $\mathcal{W}, s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an $n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm for this problem. We are not aware of any polynomial time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion planning problem involving two robots moving in a polygonal environment.

翻译：设 $\mathcal{W} \subset \mathbb{R}^2$ 为一个平面多边形环境（即可能带有空洞的多边形），顶点总数为 $n$，$A,B$ 为两个机器人，各建模为轴对齐的单位正方形，可在 $\mathcal{W}$ 内平移运动。给定 $A$ 和 $B$ 的起始与目标放置位置 $s_A,t_A,s_B,t_B \in \mathcal{W}$，目标是计算一个\emph{无碰撞运动规划} $\mathbf{\pi}^*$，即连续将 $A$ 从 $s_A$ 运动至 $t_A$、将 $B$ 从 $s_B$ 运动至 $t_B$ 的运动规划，使得在运动过程中 $A$ 和 $B$ 始终位于 $\mathcal{W}$ 内且彼此不发生碰撞。此外，若此类规划存在，我们希望返回一个最小化机器人路径总长度 $\left|\mathbf{\pi}^*\right|$ 的规划。给定 $\mathcal{W}, s_A,t_A,s_B,t_B$ 及参数 $\varepsilon > 0$，我们提出一个时间复杂度为 $n^2\varepsilon^{-O(1)} \log n$ 的 $(1+\varepsilon)$-近似算法来解决该问题。我们尚未知晓该问题的任何多项式时间算法，也不确定该问题是否为 NP-难问题。我们的结果是首个针对涉及两个在多边形环境中运动的机器人的最优运动规划问题的多项式时间 $(1+\varepsilon)$-近似算法。