We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Banyassady et al. (SoCG'22) guarantee feasibility in simple polygons under start--start and target--target distances of at least $4$, and start--target distances of at least $3$, but without optimality guarantees. Solovey et al. (RSS'15) provide a near-optimal solution in general polygonal domains, under stricter conditions: start/target positions must have pairwise distance at least $4$, and at least $\sqrt{5}\approx2.236$ from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is $2\tfrac{2}{3}$ and the obstacles-separation is $1\tfrac{2}{3}$, or (ii) the robots-separation is $\approx3.291$ and the obstacles-separation $\approx1.354$. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only $2$, and the obstacles-separation is $3$. Finally, we show that without any robots-separation assumption, obstacles-separation of at least $1.5$ may be necessary for a solution to exist.

翻译：我们研究多边形环境中单位圆盘机器人的无标签多机器人运动规划。尽管该问题在一般情况下是困难的，但在起始位置和目标位置满足适当分离假设的条件下，存在多项式时间解。Banyassady等人（SoCG'22）在起点-起点和目标-目标距离至少为$4$、起点-目标距离至少为$3$的条件下，保证简单多边形内的可行性，但未提供最优性保证。Solovey等人（RSS'15）在更严格的条件下，提供了一般多边形域中的近最优解：起始/目标位置之间的成对距离至少为$4$，且与障碍物的距离至少为$\sqrt{5}\approx2.236$。这提出了一个问题：在更密集的环境中，是否可以获得多项式时间算法？在本文中，我们提出了一种通用算法，实现了机器人分离和障碍物分离界限的不同折衷，所有这些均显著改进了现有技术水平。具体而言，我们获得了多项式时间常数近似算法，用于最小化总路径长度，当（i）机器人分离为$2\tfrac{2}{3}$且障碍物分离为$1\tfrac{2}{3}$时，或（ii）机器人分离约为$\approx3.291$且障碍物分离约为$\approx1.354$时。此外，我们引入了一种不同策略，在机器人分离仅为$2$且障碍物分离为$3$时，获得了多项式时间解。最后，我们证明，在没有任何机器人分离假设的情况下，障碍物分离至少为$1.5$可能是解存在的必要条件。