We study the problem of motion planning for a collection of $n$ labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius $2$ disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot $R$ in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As $R$ passes through a revolving area, a robot $R'$ that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time $(1+\epsilon)$-approximation algorithm. On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an $O(1)$ factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time $O(\log n \log \log n)$-approximation algorithm for this problem.

翻译：我们研究多边形环境中$n$个带标签的单位圆盘机器人的运动规划问题。假设机器人在其起点和终点附近存在旋转区域：每个起点和终点均位于自由空间中半径为$2$的圆盘内（该圆盘不必与起点或终点同心），且这些圆盘互不重叠。该假设允许一种弱单调运动规划，其中机器人按如下顺序运动：在机器人$R$的运动回合中，它从起点完整移动至终点，而其他机器人则留在各自旋转区域内。当$R$穿过某旋转区域时，位于该区域内的机器人$R'$可在区域内移动以避开碰撞。尽管存在运动规划，我们证明在此设定下（即使运动规划被限制为弱单调），最小化总移动距离问题是APX-hard的，这排除了任何多项式时间$(1+\epsilon)$近似算法的存在性。在积极方面，我们提出了首个用于计算可行弱单调运动规划的常数因子近似算法。机器人移动的总距离在最优运动规划（无需弱单调）的$O(1)$因子范围内。我们的算法可扩展至在线场景：多边形环境固定，但机器人的初始位置和最终位置以在线方式指定。最后，我们观察到在编辑路径以避免机器人碰撞时增加的总成本会因所选顺序而产生显著差异。寻找该场景下的最优顺序已知是NP-hard的，我们为此问题提供了一个多项式时间$O(\log n \log \log n)$近似算法。