Constant-Factor Time-Optimal Multi-Robot Routing on High-Dimensional Grids

Abstract

Let G = (V, E) be an m1 × . . . × mk grid for some arbitrary constant k. We establish that O(∑k i=1 mi) (makespan) time-optimal labeled (i.e., each robot has a specific goal) multi-robot path planning can be realized on G in O(|V |2 ) running time, even when vertices of G are fully occupied by robots. When all dimensions are of equal sizes, the running time approaches O(|V |). Using this base line algorithm, which provides average case O(1)-approximate (i.e., constant-factor) time-optimal solutions, we further develop a first worst case O(1)-approximate algorithm that again runs in O(|V |2 ) time for two and three dimensions. We note that the problem has a worst case running time lower bound of Ω(|V |2 ).