Minimising the longest travel distance for a group of mobile robots with interchangeable goals requires knowledge of the shortest length paths between all robots and goal destinations. Determining the exact length of the shortest paths in an environment with obstacles is NP-hard however. In this paper, we investigate when polynomial-time approximations of the shortest path search are sufficient to determine the optimal assignment of robots to goals. In particular, we propose an algorithm in which the accuracy of the path planning is iteratively increased. The approach provides a certificate when the uncertainties on estimates of the shortest paths become small enough to guarantee the optimality of the goal assignment. To this end, we apply results from assignment sensitivity assuming upper and lower bounds on the length of the shortest paths. We then provide polynomial-time methods to find such bounds by applying sampling-based path planning. The upper bounds are given by feasible paths, the lower bounds are obtained by expanding the sample set and leveraging the knowledge of the sample dispersion. We demonstrate the application of the proposed method with a multi-robot path-planning case study.

翻译：最小化一组可互换目标的移动机器人的最长行驶距离，需要掌握所有机器人与目标目的地之间的最短路径长度。然而，在存在障碍物的环境中精确确定最短路径长度是NP难的。本文研究何时多项式时间近似的最短路径搜索足以确定机器人与目标的最优分配。具体而言，我们提出一种算法，其路径规划的精度逐步提高。该算法在最短路径估计的不确定性足够小以保证目标分配最优性时提供认证。为此，我们应用基于最短路径长度上下界假设的分配敏感性理论结果。然后，我们提供多项式时间方法，通过应用基于采样的路径规划来寻找这些界。上界由可行路径给出，下界通过扩展样本集并利用样本分散度的知识获得。我们通过一个多机器人路径规划案例研究展示了所提出方法的应用。