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 $R^{\prime }$ 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.

我们研究了多边形环境中n 个标记单位圆盘机器人集合的运动规划问题。我们假设机器人在它们的起始位置和最终位置周围有旋转区域：每个起始位置和每个最终位置都包含在位于自由空间中的半径为 2 的圆盘中，不一定与起始位置或最终位置同心，它与其他起始位置无关或最终职位。这个假设允许一个弱单调的运动计划，其中机器人按照如下顺序移动：在顺序中的机器人R的转动期间，它从开始到最终位置完全移动，而其他机器人不离开它们的旋转领域。当R通过一个旋转区域时，一个机器人$R^{‘}$位于该区域内的可以在旋转区域内移动以避免碰撞。尽管存在运动计划，但我们表明，在此设置中最小化总行进距离，特别是即使运动计划被限制为弱单调时，也是 APX 困难的，排除了任何多项式时间$(\mathrm{1个}+\epsilon )$-近似算法。

在积极的一面，我们提出了第一个用于计算可行的弱单调运动计划的常数因子近似算法。机器人行进的总距离在$欧(\mathrm{1个})$最佳运动计划的因素，它不需要是弱单调的。我们的算法扩展到在线设置，其中多边形环境是固定的，但机器人的初始和最终位置以在线方式指定。最后，我们观察到，根据我们选择的顺序，我们在编辑路径以避免机器人碰撞时增加的总成本开销可能会有很大差异。已知在这方面找到最佳排序是 NP 难的，我们提供多项式时间$欧(\mathrm{日志}n\mathrm{日志}\mathrm{日志}n)$-这个问题的近似算法。