Given a set of $n$ point robots inside a simple polygon $P$, the task is to move the robots from their starting positions to their target positions along their shortest paths, while the mutual visibility of these robots is preserved. Previous work only considered two robots. In this paper, we present an $O(mn)$ time algorithm, where $m$ is the complexity of the polygon, when all the starting positions lie on a line segment $S$, all the target positions lie on a line segment $T$, and $S$ and $T$ do not intersect. We also argue that there is no polynomial-time algorithm, whose running time depends only on $n$ and $m$, that uses a single strategy for the case where $S$ and $T$ intersect.

翻译：给定一个简单多边形 $P$ 内部的 $n$ 个点机器人，任务是在保持这些机器人相互可见性的前提下，将机器人从起始位置沿其各自的最短路径移动到目标位置。先前的工作仅考虑了两个机器人。本文提出了一种运行时间为 $O(mn)$ 的算法，其中 $m$ 为多边形的复杂度，适用于所有起始位置位于线段 $S$ 上、所有目标位置位于线段 $T$ 上，且 $S$ 与 $T$ 不相交的情况。此外，我们论证了对于 $S$ 与 $T$ 相交的情况，不存在仅依赖于 $n$ 和 $m$ 且使用单一策略的多项式时间算法。