Fast algorithms for optimal multi-robot path planning are sought after in many real-world applications. Known methods, however, generally do not simultaneously guarantee good solution optimality and fast run time (e.g., polynomial). In this work, we develop a low-polynomial running time algorithm, called SplitAndGroup (SAG),that solves the multi-robot path planning problem on grids and grid-like environments and produces constant factor makespan-optimal solutions in the average case. That is, SAG is an average case O(1)-approximation algorithm. SAG computes solutions with sub-linear makespan and is capable of handling cases when the density of robots is extremely high - in a graph-theoretic setting, the algorithm supports cases where all vertices of the underlying graph are occupied by robots. SAG attains its desirable properties through a careful combination of divide-and-conquer technique and network flow based methods for routing the robots. Solutions from SAG, in a weaker sense, is also a constant factor approximation on total distance optimality.
View on arXiv