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

TL;DR

This paper tackles the pebble motion on trees (PMT) within the broader multi-agent path finding (MAPF) framework by introducing two practical solvers with improved length complexity. The CATERPILLAR algorithm solves motion planning on trees with a bound of $O(n c)$ moves by constructing caterpillar sets that partition the path between source and target and facilitate obstacle clearance. The Leaves procedure solves PMT with a bound of $O(k n c + n^2)$ by reducing to unlabeled PMT to generate intermediate targets and then solving $|P|$ motion-planning problems on subtrees before applying the inverse of the unlabeled solution. Additionally, the ts-PMT variant handles trans-shipment vertices (nodes that cannot host pebbles) and provides modifications to all components, enabling reductions from general MAPF on graphs. Experimental results on random trees corroborate the theoretical bounds and demonstrate scalable performance, highlighting practical implications for suboptimal MAPF solvers on trees and for reductions to general graphs via ts-PMT.

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.

Figures (13)

• Figure 1: Example of Bring hole from $w$ to $v$. Green squares represent pebbles, blue circles represent holes.
• Figure 2: Example of Move Pebble from $v$ to $w$. Green squares represent pebbles, blue circles represent holes.
• Figure 3: Example of Gather hole problem. We want to move three closest holes to the subtree $\bar{T}$.
• Figure 4: We consider the motion planning problem with source vertex $r$ and target vertex $t$ on a tree with $c=5$. $S_0$, $S_1$ and $S_2$ are the caterpillar sets along path $\pi_{rt}$.
• Figure 5: Example of execution of an iteration of the for cycle of Step 3 of Procedure A. Blue circles are holes, green squares are obstacles and the red square is the marked pebble.

Theorems & Definitions (19)

• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Proposition 3.2
• Remark 4.1
• Proposition 4.3
• proof