Optimal Unlabeled Pebble Motion on Trees and its Application to Multi-Agent Path Finding

TL;DR

This paper introduces an optimal linear-time algorithm for the Unlabeled Pebble Motion on Trees problem, enabling efficient solutions for multi-agent pathfinding with proven bounds on various optimality criteria.

Contribution

It presents the first asymptotically optimal algorithm for UPMT and extends it to solve unlabeled MAPF on trees, with new bounds on optimal metrics.

Findings

Algorithm runs in linear time relative to input size.

Provides bounds on makespan, sum of costs, and plan length for MAPF.

Achieves optimal solutions for pebble motion and multi-agent pathfinding.

Abstract

Given a tree, a set of pebbles initially stationed at some nodes of the tree, and a set of target nodes, the Unlabeled Pebble Motion on Trees problem (UPMT) asks to find a plan to move the pebbles one-at-a-time from the starting nodes to the target nodes along the edges of the tree while minimizing the number of moves. This paper proposes the first optimal algorithm for UPMT that is asymptotically as fast as possible, as it runs in a time linear in the size of the input (the tree) and the size of the output (the optimal plan). We extend this to solve unlabeled Multi-Agent Path Finding (MAPF) in trees, providing novel bounds on optimal makespan, sum of costs, and pebble motion plan length.