Polynomial-time Configuration Generator for Connected Unlabeled Multi-Agent Pathfinding

TL;DR

This work tackles connected unlabeled multi-agent pathfinding (CUMAPF), where agents must move to target placements while preserving global connectivity, a problem known to be NP-hard. It introduces PULL, a complete, suboptimal configuration generator with a theoretical per-step complexity of $O(\Delta^2 n^2)$ and a constructive method to build connected, collision-free next configurations by pulling agents along multiple vertex-disjoint paths toward targets. The authors prove PULL’s completeness, analyze its upper bound on makespan, and reveal both tight and adversarial instances, highlighting limits and guiding when to deploy it. To bolster performance on large or challenging cases, they integrate PULL into LaCAM$^*$, yielding an eventually optimal solver, and present extensive empirical evidence showing competitive solution quality and favorable scaling on hundreds of agents across diverse maps. The results have practical implications for swarm robotics and programmable matter where maintaining connectivity is essential during reconfiguration and marching tasks.

Abstract

We consider Connected Unlabeled Multi-Agent Pathfinding (CUMAPF), a variant of MAPF where the agents must maintain connectivity at all times. This problem is fundamental to swarm robotics applications like self-reconfiguration and marching, where standard MAPF is insufficient as it does not guarantee the required connectivity between agents. While unlabeled MAPF is tractable in optimization, CUMAPF is NP-hard even on highly restricted graph classes. To tackle this challenge, we propose PULL, a complete and polynomial-time algorithm with a simple design. It is based on a rule-based one-step function that computes a subsequent configuration that preserves connectivity and advances towards the target configuration. PULL is lightweight, and runs in $O(n^2)$ time per step in 2D grid, where $n$ is the number of agents. Our experiments further demonstrate its practical performance: PULL finds competitive solution qualities against trivial solutions for hundreds of agents, in randomly generated instances. Furthermore, we develop an eventually optimal solver that integrates PULL into an existing search-based MAPF algorithm, providing a valuable tool for small-scale instances.

Figures (11)

• Figure 1: The naive approach for solving CUMAPF. Light red represents $S=\mathcal{Q}_{0}$, deep red represents the current configuration, and blue represents $T$. (Left) Initial configuration, (Center) intermediate configuration at step $t<t'$, (Right) configuration at step $t'=\mathop{\mathrm{\textsf{dist}}}\limits(x,y)$.
• Figure 2: An example of the PULL's operation. The graph $G$ is 5$\times$3 grid, and $v_{\texttt{i-j}}$ denotes the vertex with row $i$, column $j$. Light red indicates $\mathcal{Q}^{\text{from}}$, and blue represents $T$. (Column 1) first PULL call ${\texttt{PULL}}\xspace(v_{\texttt{4-3}}, \emptyset,\{v_{\texttt{3-2}},v_{\texttt{4-1}},v_{\texttt{4-2}}\})$ ($v_{\texttt{4-3}}$ is marked by yellow). Then $V_S=\{v_{\texttt{1-2}},v_{\texttt{2-1}},v_{\texttt{2-2}},v_{\texttt{2-3}}\}$, and $\textsf{cur}=v_{\texttt{1-2}}$. PULL returns $R = \{0,2,4,6\}$. (Column 2) PULL call ${\texttt{PULL}}\xspace(v_{\texttt{5-1}}, R,\{v_{\texttt{3-2}},v_{\texttt{4-1}},v_{\texttt{4-2}}\})$. Then $V_S=\emptyset$, and ${\texttt{PULL}}\xspace$ return $R$. The call ${\texttt{PULL}}\xspace$ with $t=v_{\texttt{5-2}}$ also fails. We set $R= \{0,2,4,5,6\}$. (Column 3) PULL calls ${\texttt{PULL}}\xspace(v_{\texttt{3-1}},R, \emptyset)$. $V_S=\{v_{\texttt{2-1}}\}$, and PULL returns $R=\{0,1,2,4,5,6\}$. (Column 4) Repeatedly calling PULL, finally PULL returns $\mathcal{Q}^{\text{to}}=[v_{\texttt{2-2}},v_{\texttt{3-1}},v_{\texttt{3-2}},v_{\texttt{3-3}},v_{\texttt{4-2}},v_{\texttt{4-1}},v_{\texttt{4-3}}]$.
• Figure 3: A conceptual illustration of line \ref{['pull:sortcc']}. In the initial iteration, PULL calls PULL on $N(P_0)$, which increases the size of the largest component. (Left) initial configuration, (Right) itermediate configuration before executing line \ref{['pull:notg_s']}. Two connected components might merge (e.g., $P_0$ and $P_1$ in the figure), and these are then computed as a larger component in the next step.
• Figure 4: A brief explanation of the proof in \ref{['lem: firstcall']}. (Left) Graph $G[\mathcal{Q}^{\text{from}} \cup \{t_{1}\} \setminus \{w\}]$ is divided into multiple components, shown as two for convenience. (Right) A graph $G[\mathcal{Q}^{\text{from}} \cup \{t_{1}\}]$ has a single component, and every $(v',u')$-paths must pass through $w$ since $w$ is a cut vertex.
• Figure 5: (a) An example of a tight instance: the optimal makespan is $\mathop{\mathrm{\textsf{diam}}}\limits(G)+|A|-1$. (b) An example of a adversarial instance: PULL outputs plan with $O(|A|)$ makespan, while the optimal makespan is two.

Theorems & Definitions (23)

• Lemma 0: $\ast$
• proof
• Lemma 1
• proof
• Lemma 1: $\ast$
• Lemma 1: $\ast$
• Theorem 2
• Proposition 3
• Proposition 3: $\ast$
• Corollary 4