Abstract Path Planning for Multiple Robots: A Theoretical Study

An abstraction of the problem of multi-robot path planning is introduced in this paper. The basic task is to determine spatial-temporal plan for each robot of a group of robots where each robot is given its initial position in the environment and it needs to go to the given goal position. Robots must avoid obstacles and must not collide with each other. The abstraction adopted in this work models the environment within that robots are moving as an undirected graph. Robots are placed in vertices of the graph; at most one robot is placed in each vertex and at most one vertex remains unoccupied .The move is allowed into the unoccupied vertex or into the vertex being vacated by an allowed move supposed that no other robot is entering the same target vertex. The relation of multirobot path planning to the problem of pebble motion on a graph (which the most widely known representative is 15-puzzle) is discussed. The optimization variant of the abstract multi-robot path planning is particularly studied. The task is to find a solution of the makespan as small as possible in the optimization variant. The main contribution of the paper is the proof of the NP-completeness of the decision version of the optimization variant of multi-robot path planning. The reduction of Boolean satisfiability to multi-robot path planning is used in the proof.