Solvability of multi robot motion planning problems on Trees

Multi robot motion planning problems can be solved very efficiently when the configuration space is mapped onto a graph. Before planning, however, it must be assured that the constructed graph is reachable (solvable) for the given number and configuration of robots. Solvable trees are types of trees that let any arrangement of a specified number of robots be reached from any initial arrangement through sequential moves of robots on vertices of the tree. In this paper the properties of solvable trees are investigated, and minimal solvable trees, which are the smallest solvable trees in terms of the number of vertices, are introduced as a new concept. Also, a new algorithm with linear time complexity is proposed for deciding whether a multi robot motion planning problem has a solution on a tree, without explicitly solving it.