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Fast algorithms for optimal multi-robot path planning are sought after in real-world applications. Known methods, however, generally do not simultaneously guarantee good solution optimality and good (e.g., polynomial) running time. In this work, we develop a first 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 on average over all problem instances. That is, SaG is an average case O (1)-approximation algorithm and computes solutions with sub-linear makespan. SaG 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. SaG attains its desirable properties through a careful combination of a novel divide-and-conquer technique, which we denote as global decoupling , and network flow based methods for routing the robots. Solutions from SaG , in a weaker sense, are also a constant factor approximation on total distance optimality.

Average case constant factor time and distance optimal multi-robot path planning in well-connected environments