An Algorithm with Improved Complexity for Pebble Motion/Multi-Agent Path Finding on Trees

Abstract

The pebble motion on trees (PMT) problem consists in finding a feasible sequence of moves that repositions a set of pebbles to assigned target vertices. This problem has been widely studied because, in many cases, the more general Multi-Agent path finding (MAPF) problem on graphs can be reduced to PMT. We propose a simple and easy to implement procedure, which finds solutions of length O(|P|nc + n²), where n is the number of nodes, P is the set of pebbles, and c the maximum length of corridors in the tree. This complexity result is more detailed than the current best known result O(n³), which is equal to our result in the worst case, but does not capture the dependency on c and |P|.