Certification of Bottleneck Task Assignment with Shortest Path Criteria

TL;DR

This paper presents a method for certifying optimal robot-to-goal assignments by iteratively improving shortest path estimates using sampling-based planning, ensuring polynomial-time solutions with guarantees.

Contribution

It introduces an algorithm that certifies optimality of robot assignments using bounds on shortest paths, combining sampling-based planning with assignment sensitivity analysis.

Findings

The method guarantees optimal assignment when path estimate uncertainties are sufficiently small.

Sampling-based bounds can be computed efficiently to certify optimality.

Application to multi-robot path planning demonstrates practical effectiveness.

Abstract

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.