Coordinated Motion Planning: Multi-Agent Path Finding in a Densely Packed, Bounded Domain

TL;DR

This work extends multi-agent path finding to densely packed agents within simple polyomino domains, addressing feasibility and efficiency under strict boundary constraints. It provides a full characterization of universal reconfigurability through a cover by $2\times 2$ squares with a connected intersection graph and introduces the bottleneck $\zeta_{\text{bottleneck}}(P)$ and depth $\mu_{\text{depth}}(P)$ as core shape parameters that govern makespan and stretch. The authors derive both lower and upper bounds that connect domain geometry to performance, including a lower bound of $\Omega\big(d_{\text{diameter}} + d_{\text{diameter}}^2 / \zeta_{\text{bottleneck}}(P)\big)$ and constructive schedules achieving near-optimal bounds via permutation routing and local swap techniques, particularly in scaled or narrow instances. They show that non-convex boundaries can force significant increases in makespan, while scalable tiling and RotateSort-based strategies yield efficient schedules, generalizing prior results on rectangles and convex grid pieces to broader polyomino domains. The results have practical implications for constrained-grid robotics and warehousing, where dense, coordinated motion is required within bounded, nontrivial geometries.

Abstract

We study Multi-Agent Path Finding for arrangements of labeled agents in the interior of a simply connected domain: Given a unique start and target position for each agent, the goal is to find a sequence of parallel, collision-free agent motions that minimizes the overall time (the makespan) until all agents have reached their respective targets. A natural case is that of a simply connected polygonal domain with axis-parallel boundaries and integer coordinates, i.e., a simple polyomino, which amounts to a simply connected union of lattice unit squares or cells. We focus on the particularly challenging setting of densely packed agents, i.e., one per cell, which strongly restricts the mobility of agents, and requires intricate coordination of motion. We provide a variety of novel results for this problem, including (1) a characterization of polyominoes in which a reconfiguration plan is guaranteed to exist; (2) a characterization of shape parameters that induce worst-case bounds on the makespan; (3) a suite of algorithms to achieve asymptotically worst-case optimal performance with respect to the achievable stretch for cases with severely limited maneuverability. This corresponds to bounding the ratio between obtained makespan and the lower bound provided by the max-min distance between the start and target position of any agent and our shape parameters. Our results extend findings by Demaine et al. (SIAM Journal on Computing, 2019) who investigated the problem for solid rectangular domains, and in the closely related field of Permutation Routing, as presented by Alpert et al. (Computational Geometry, 2022) for convex pieces of grid graphs.

Figures (5)

• Figure 1: A polyomino, its dual graph, and a cut. Subsequent illustrations will only show the boundary and any relevant cuts, foregoing the underlying integer grid.
• Figure 2: An illustration of an configurations and a transformations.
• Figure 3: In polyominoes composed of two $2\times 2$ squares, we can realize swaps in $\mathcal{O}(1)$ steps.
• Figure 4: We illustrate a class of instances which require $\Omega(d diameter\xspace^2)$ transformations.
• Figure 6: An illustration of a scaled polyomino $P$, its tiles, and their corresponding dual graph.

Theorems & Definitions (10)

• Theorem 1
• Lemma 2
• Lemma 4
• Proposition 5
• Proposition 6
• Lemma 7
• Theorem 8
• Proposition 9
• Lemma 10
• Lemma 11