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

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.

Figures (10)

• Figure 1: Example configuration of nodes labeled with demands. Green diamonds represent pebbles and red squares represent targets.
• Figure 2: Example configuration of nodes, labeled $A - G$ in order of processing. The final values of $s(u)$ and $l(u)$ are shown. The agent at $A$ travels the path $A - B - D - F - G$ with no initial wait. The agent at $C$ waits at timesteps $0$ and $1$, giving way to the agent at $A$. The agent at $E$ waits at timesteps $0$, $1$ and $2$, giving way to the agent at $A$.
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• Figure 5: Example configuration where the bounds in Theorem \ref{['thm:makespan-upper-bound']} and corollaries \ref{['thm:sum-of-costs-upper-bound']} and \ref{['thm:OPT-upper-k-n-k']} are tight.

Theorems & Definitions (94)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• Lemma 6