An algorithm with improved complexity for pebble motion/multi-agent path finding on trees

TL;DR

This paper introduces a new algorithm for pebble motion on trees that improves complexity analysis by explicitly incorporating parameters like pebbles, nodes, and corridor length, offering more precise performance insights.

Contribution

It presents a simple, implementable algorithm with complexity O(knc + n^2), providing a more detailed analysis than previous worst-case bounds.

Findings

Solution length is O(knc + n^2)

Complexity depends on pebbles, nodes, and corridor length

Improves upon the previous O(n^3) bound

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(knc + n^2), where n is the number of nodes, $k$ is the number 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(n^3), which is equal to our result in the worst case, but does not capture the dependency on c and k.