Efficiently Reconfiguring a Connected Swarm of Labeled Robots

TL;DR

The paper tackles connected, labeled multi-robot reconfiguration on a grid, seeking minimal makespan while preserving connectivity. It proves a fundamental $\Omega(\sqrt{n})$ lower bound on stretch for general connected labeled reconfiguration and, in contrast, shows constant-stretch schedules are achievable for scaled (highly aggregated) configurations. It also establishes NP-completeness for deciding makespan $2$ in the labeled connected setting, with polynomial-time verifiability for makespan $1$. The authors introduce a tiling-based scaffold and a phase-driven algorithm that enables efficient, parallel, connectivity-preserving reconfiguration via local interior/boundary moves and inter-tile exchanges, culminating in a constant-stretch schedule for sufficiently large scales. These results advance understanding of connected coordination in dense robot swarms and provide a scalable framework for practical reconfiguration in automated systems.

Abstract

When considering motion planning for a swarm of $n$ labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, collision-free robot motions. The objective is to reach the new configuration in a minimum amount of time; an important constraint is to keep the swarm connected at all times. Problems of this type have been considered before, with recent notable results achieving constant stretch for not necessarily connected reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, the total duration of an overall schedule can be bounded to $\mathcal{O}(d)$, which is optimal up to constant factors. However, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations (which arise by increasing all dimensions of a given object by the same multiplicative factor) of unlabeled robots. We resolve these major open problems by (1) establishing a lower bound of $Ω(\sqrt{n})$ for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected reconfiguration can be achieved. In addition, we show that (3) it is NP-complete to decide whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.

Figures (9)

• Figure 1: Overview of the structure used for the reduction.
• Figure 2: The separation gadget.
• Figure 3: A variable gadget and schedules corresponding to assignments of $0$ and $1$.
• Figure 5: Pairs of robots must swap their positions (a), which can only be realized by schedules using moves that involve all robots (b).
• Figure 6: Tilings and tiled configurations. In (a), we see a configuration, a $7$-tiling, and its boundary regions. A valid tiled configuration can be seen in (b), alongside an invalid one in (c) with a missing tile boundary highlighted in orange.

Theorems & Definitions (13)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Theorem 5
• Lemma 6: RotateSort
• Lemma 7
• Theorem 8
• Corollary 9
• Lemma 10