Certification of Bottleneck Task Assignment with Shortest Path Criteria

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.