Connected Unlabeled MAPF (CUMAPF)

• CUMAPF is a graph-based planning problem where interchangeable agents move from an initial set to a target set, maintaining a connected occupied subgraph at every time step.
• The added connectivity constraint makes the problem NP-hard even on restricted graph classes, contrasting with standard and unlabeled MAPF variants.
• The PULL algorithm offers a complete, polynomial-time configuration generator that preserves connectivity, albeit with trade-offs in global optimality.

Searching arXiv for CUMAPF and closely related connected MAPF papers. Connected Unlabeled Multi-Agent Pathfinding (CUMAPF) is a variant of multi-agent pathfinding in which a team of identical agents must move from an initial occupied vertex set to a target vertex set while avoiding conflicts and remaining connected as a group at every time step. In the graph-based formulation emphasized by recent work, the agents are interchangeable, the occupied vertices at each step must induce a connected subgraph, and the objective is to minimize makespan (Suzuki et al., 22 Oct 2025). CUMAPF is distinguished both from standard MAPF, which does not require team connectivity, and from unlabeled MAPF, which is polynomial-time solvable in optimization form via max-flow reductions, because the added connectivity constraint makes the problem substantially harder; recent work explicitly notes NP-hardness even on highly restricted graph classes, including 2D grid settings (Suzuki et al., 22 Oct 2025).

1. Formal definition and problem scope

The standard graph model used for CUMAPF takes a finite, simple, connected, undirected graph $G=(V,E)$, a set of $n$ agents, and initial and target vertex sets $S,T\subseteq V$ with $|S|=|T|=n$. A configuration is written as

$\mathcal Q=(q_1,\dots,q_n)\in V^n,$

where $q_i$ is agent $i$'s vertex, and a configuration is connected if the induced subgraph $G[\mathcal Q]$ is connected (Suzuki et al., 22 Oct 2025).

A plan is a sequence

$\Pi=[Q_0=S,Q_1,\dots,Q_{t^\ast}=T]$

subject to four conditions. First, it must be reachable: each agent moves to a neighbor or waits. Second, it must be vertex-conflict-free. Third, it must be swap-conflict-free. Fourth, it must be connected at every step, meaning every intermediate configuration induces a connected subgraph (Suzuki et al., 22 Oct 2025). The objective is to minimize makespan $t^\ast$ (Suzuki et al., 22 Oct 2025).

This formulation makes the unlabeled character explicit: agents are interchangeable, and the target specification is a set rather than an agent-specific assignment. That unlabeled aspect contrasts with labeled connected MAPF formulations in which each agent $n$0 has its own designated goal $n$1. A closely related but distinct example is MALCR, “Multi-Agent Path Finding with Limited Communication Range,” where each agent has a specific start and a specific goal and the team must remain connected through communication links at all times (Bui et al., 6 Jan 2025). That problem is structurally close to the “connected” component of CUMAPF, but it is not a standard unlabeled formulation (Bui et al., 6 Jan 2025).

2. Connectivity as the defining constraint

In CUMAPF, connectivity is enforced configuration-wise: every intermediate configuration $n$2 must induce a connected subgraph $n$3 (Suzuki et al., 22 Oct 2025). This requirement is global rather than pairwise. Standard MAPF algorithms typically reason about local collision constraints, but CUMAPF requires preservation of a connected occupied set throughout the entire execution (Suzuki et al., 22 Oct 2025).

Recent algorithmic work enforces this constraint by restricting which occupied vertices may be vacated. In PULL, the next configuration is generated by moving agents along a single path-like chain of assignments while ensuring that no moved vertex is a cut vertex of the occupied subgraph after the move (Suzuki et al., 22 Oct 2025). The paper formalizes this using $n$4, the set of vertices reachable from $n$5, and $n$6, the set of cut vertices. For a candidate starting vertex $n$7, the safe starting set is

$n$8

where $n$9 is a reachable set and $S,T\subseteq V$0 is the set of cut vertices in the relevant induced graph (Suzuki et al., 22 Oct 2025). The central invariant is stated explicitly:

If the current configuration is connected, then after each internal assignment step the new configuration remains connected, collision-free, and reachable. (Suzuki et al., 22 Oct 2025)

This connectivity notion differs from communication-constrained connected MAPF formulations. In MALCR, pairwise communication is governed by an agent communication constraint (ACOMM), under which two agents can communicate only if their Euclidean distance is within a range $S,T\subseteq V$1, and the full team must satisfy a team communication constraint (TCOMM) by forming a spanning tree whose edges satisfy ACOMM (Bui et al., 6 Jan 2025). That is an always-connected communication graph under a range bound, rather than the induced-subgraph connectivity condition used in graph-based CUMAPF (Bui et al., 6 Jan 2025).

A second distinction arises in decentralized unlabeled planning. TP-SWAP assumes a connected underlying graph and local communication neighborhoods, with chain communication allowed, but it does not require a single global connected communication graph at all times (Dergachev et al., 2024). This makes it relevant to distributed coordination in CUMAPF-like settings, but not identical to formulations that require continuous global connectivity of the full team (Dergachev et al., 2024).

3. Complexity and hardness

The complexity-theoretic position of CUMAPF is one of its defining features. While unlabeled MAPF is polynomial-time solvable for the optimization version via max-flow reductions, CUMAPF is reported as NP-hard even on very restricted graphs (Suzuki et al., 22 Oct 2025). The same work notes two stronger hardness statements: deciding whether the optimal makespan is $S,T\subseteq V$2 or more is NP-hard, and approximating makespan within factor $S,T\subseteq V$3 is NP-hard (Suzuki et al., 22 Oct 2025).

These hardness results motivate the search for algorithms that are complete and polynomial-time per step, or exact only on small instances. PULL is presented as a complete and polynomial-time algorithm with a simple design, operating as a rule-based one-step configuration generator (Suzuki et al., 22 Oct 2025). Its per-step runtime is

$S,T\subseteq V$4

where $S,T\subseteq V$5 is the maximum degree of the graph, and in 2D grid graphs this becomes

$S,T\subseteq V$6

because $S,T\subseteq V$7 is constant (Suzuki et al., 22 Oct 2025). Using the stated makespan bound,

$S,T\subseteq V$8

the full-plan runtime is

$S,T\subseteq V$9

and the bound is stated as tight (Suzuki et al., 22 Oct 2025).

The same paper also makes clear that polynomial-time completeness does not imply global optimality. It gives adversarial instances where PULL produces a makespan $|S|=|T|=n$0 even when the optimum is $|S|=|T|=n$1 (Suzuki et al., 22 Oct 2025). This establishes a central structural fact about CUMAPF: tractable complete generation of connected plans is possible, but exact optimality remains difficult enough to require heavier search procedures.

4. The PULL configuration generator

PULL is the first algorithm in the supplied material that directly targets CUMAPF as such (Suzuki et al., 22 Oct 2025). It is framed as a configuration generator: given the current connected configuration $|S|=|T|=n$2 and target set $|S|=|T|=n$3, it outputs a subsequent connected configuration $|S|=|T|=n$4 in one step (Suzuki et al., 22 Oct 2025). Repeated application from $|S|=|T|=n$5 eventually reaches $|S|=|T|=n$6 (Suzuki et al., 22 Oct 2025).

Its high-level behavior has two cases. If some target vertices are already occupied, meaning $|S|=|T|=n$7, the algorithm finds connected components of $|S|=|T|=n$8, sorts them by size, and processes neighboring target vertices around the largest components first so that the occupied target region grows monotonically (Suzuki et al., 22 Oct 2025). Otherwise, when $|S|=|T|=n$9, it sorts neighboring candidate vertices by their distance to the target set and pulls one agent along a shortest route toward the target (Suzuki et al., 22 Oct 2025).

The central inner procedure is written as $\mathcal Q=(q_1,\dots,q_n)\in V^n,$0, where $\mathcal Q=(q_1,\dots,q_n)\in V^n,$1 is a candidate vertex to grow toward, $\mathcal Q=(q_1,\dots,q_n)\in V^n,$2 is the set of agents already committed in the current one-step generation, and $\mathcal Q=(q_1,\dots,q_n)\in V^n,$3 is a set of vertices forbidden as starting points for the current chain (Suzuki et al., 22 Oct 2025). It computes reachable vertices $\mathcal Q=(q_1,\dots,q_n)\in V^n,$4, cut vertices $\mathcal Q=(q_1,\dots,q_n)\in V^n,$5, then forms

$\mathcal Q=(q_1,\dots,q_n)\in V^n,$6

If $\mathcal Q=(q_1,\dots,q_n)\in V^n,$7, the procedure fails. Otherwise it selects

$\mathcal Q=(q_1,\dots,q_n)\in V^n,$8

and repeatedly assigns the agent at $\mathcal Q=(q_1,\dots,q_n)\in V^n,$9 to move one step closer to $q_i$0 along a BFS tree,

$q_i$1

until the chain reaches $q_i$2 or no safe vertex remains (Suzuki et al., 22 Oct 2025).

The completeness proof relies on two monotonic progress arguments. If initially $q_i$3, each successful step decreases the minimum distance between some occupied vertex and the target by $q_i$4. Once $q_i$5, the size of the largest connected component of $q_i$6 grows monotonically, with

$q_i$7

Therefore, after finitely many steps, all agents occupy target vertices (Suzuki et al., 22 Oct 2025).

The following table summarizes the main stated properties of PULL.

Aspect	Stated property	Source
Problem class	Connected Unlabeled Multi-Agent Pathfinding	(Suzuki et al., 22 Oct 2025)
Algorithm type	Complete, polynomial-time, rule-based configuration generator	(Suzuki et al., 22 Oct 2025)
Per-step runtime	$q_i$8; $q_i$9 in 2D grid	(Suzuki et al., 22 Oct 2025)
Makespan bound	$i$0	(Suzuki et al., 22 Oct 2025)
Optimality status	Complete but not globally optimal	(Suzuki et al., 22 Oct 2025)

A plausible implication is that PULL is best viewed as a constructive backbone for CUMAPF: it guarantees connected progress under the model assumptions, but its deterministic rule structure trades off optimality for completeness and scalability.

5. Exact solving and hybrid search

Because PULL can be suboptimal, the same work integrates it with LaCAM*, a search-based MAPF solver that is eventually optimal (Suzuki et al., 22 Oct 2025). The adapted procedure takes the current configuration $i$1, targets $i$2, and a set of search constraints $i$3, where the constraints specify certain agents and their required next-step moves (Suzuki et al., 22 Oct 2025). If a constrained agent conflicts with an unconstrained one, the algorithm raises the priority of the relevant vertex so that PULL will preferentially move the conflicting agent away (Suzuki et al., 22 Oct 2025). After generation, the result is checked, and if it is disconnected or still contains vertex conflicts, the procedure returns $i$4; swap conflicts are then resolved or rejected (Suzuki et al., 22 Oct 2025).

In this context, “eventually optimal” means that as the search continues, the solver eventually finds and returns an optimal makespan solution, provided enough time (Suzuki et al., 22 Oct 2025). The experiments reported for this combined solver use adversarial small instances such as 2-3-0, 5-5-0, and 10-3-0, where it can recover optimal solutions on sufficiently small cases, but may struggle to reach optimality within a two-hour budget on harder instances (Suzuki et al., 22 Oct 2025). This suggests that exact CUMAPF remains computationally expensive even when a complete configuration generator is available.

A related exact-search perspective appears in connected labeled MAPF. CCBS adapts conflict-based search to Connected Multi-Agent Path Finding (CMAPF), replacing collision conflicts with disconnection conflicts relative to a designated base (Queffelec et al., 2020). In that model, a configuration is connected iff the set $i$5 forms a connected subgraph under the communication relation $i$6, and the optimization objective is makespan (Queffelec et al., 2020). The Neg-only variant is stated to be complete and optimal, while the Self/Other-only variant is not complete in general, though it is complete in a special case on sight-moveable graphs when all agents start at the base (Queffelec et al., 2020). Although this is not an unlabeled formulation and the core CMAPF model allows co-location rather than enforcing standard MAPF collision rules, it provides an exact-search template for connectivity-constrained planning (Queffelec et al., 2020).

This suggests a broader methodological pattern: CUMAPF admits lightweight complete generation procedures such as PULL, but exact solving typically requires embedding connectivity-aware generation into a larger search framework.

6. Relations to neighboring formulations

CUMAPF sits at the intersection of three neighboring problem classes: unlabeled MAPF, connected MAPF, and communication-constrained multi-agent planning.

The relationship to unlabeled MAPF is direct. In AMAPF, agents are interchangeable and the requirement is that each goal be reached by some agent and every agent end at some goal (Dergachev et al., 2024). TP-SWAP addresses this setting in a fully decentralized fashion by combining target swapping, priority swapping, and a local rule-based execution procedure (Dergachev et al., 2024). Each agent maintains a target-priority table $i$7, where $i$8 stores the highest priority currently known to have claimed target $i$9, and uses TP-UPDATE to restore assignment consistency within local communication subgroups (Dergachev et al., 2024). The paper proves completeness under assumptions including a connected graph $G[\mathcal Q]$0, instantaneous and error-free communication, synchronized movement, local communication subgroups, unique priorities, and a finite number of targets equal to the number of agents (Dergachev et al., 2024).

The relationship to connected MAPF is more nuanced. CCBS studies a variant in which agents must remain connected to each other and to a designated base throughout execution (Queffelec et al., 2020). That is a connectivity requirement of the same general family, but the model differs in three important respects recorded in the supplied material: it is labeled, it optimizes over executions in a topological graph $G[\mathcal Q]$1, and the core formulation explicitly allows agents to share vertices at the same time step (Queffelec et al., 2020). It is therefore relevant as a connectivity-search method rather than as a direct solver for unlabeled collision-avoiding CUMAPF.

The relationship to communication-constrained planning is illustrated by MALCR. There, connectivity is maintained through a communication-range constraint rather than through connectivity of the occupied subgraph itself (Bui et al., 6 Jan 2025). The team must satisfy TCOMM, meaning a spanning tree over agents whose edges satisfy the pairwise distance condition of ACOMM (Bui et al., 6 Jan 2025). The paper’s major contribution is dynamic leading, implemented in the MA-DL framework, where the leader is not fixed globally and may be reselected when planning stalls or when communication-at-goal issues arise (Bui et al., 6 Jan 2025). The method is explicitly labeled, not unlabeled, but it is highly relevant to the connectivity-maintenance side of CUMAPF (Bui et al., 6 Jan 2025).

A concise comparison is useful.

Formulation	Unlabeled?	Connectivity requirement
CUMAPF / PULL	Yes	$G[\mathcal Q]$2 connected at every step
TP-SWAP AMAPF	Yes	Connected workspace graph; local communication subgroups
MALCR / MA-DL	No	Team connected by communication links under range $G[\mathcal Q]$3
CMAPF / CCBS	No	Agents connected to each other and to a designated base

This taxonomy clarifies a common misconception. Not every “connected MAPF” result is a CUMAPF result. MALCR and CCBS address connectivity, but neither is an unlabeled formulation (Bui et al., 6 Jan 2025, Queffelec et al., 2020). TP-SWAP is unlabeled and decentralized, but it does not require continuous global connectivity among all agents (Dergachev et al., 2024). PULL is the direct CUMAPF algorithm in the supplied set (Suzuki et al., 22 Oct 2025).

7. Empirical behavior, applications, and limitations

The experimental profile reported for PULL emphasizes scalability on randomly generated instances with hundreds of agents (Suzuki et al., 22 Oct 2025). The experiments use random-32-32-20, random-48-48-20, random-64-64-20, and warehouse-10-20-10-2-2; they generate instances with $G[\mathcal Q]$4 to $G[\mathcal Q]$5 agents and two random connected subgraphs as initial and target sets (Suzuki et al., 22 Oct 2025). The implementation is in Python and was run on a Mini PC with Intel Core i9-13900H CPU and 32 GB RAM (Suzuki et al., 22 Oct 2025). Since there was no prior suboptimal CUMAPF solver, the baseline is a simple “single-path” variant, together with a trivial lower bound based on bottleneck matching $G[\mathcal Q]$6 (Suzuki et al., 22 Oct 2025). The paper reports that runtime scales well and grows more slowly than the worst-case $G[\mathcal Q]$7 trend in practice; in 2D grids, the average time depends mainly on $G[\mathcal Q]$8, not strongly on $G[\mathcal Q]$9, matching the theory (Suzuki et al., 22 Oct 2025). It also reports that PULL’s makespan relative to $\Pi=[Q_0=S,Q_1,\dots,Q_{t^\ast}=T]$0 is substantially smaller than the simple baseline, and for $\Pi=[Q_0=S,Q_1,\dots,Q_{t^\ast}=T]$1, PULL’s makespan/$\Pi=[Q_0=S,Q_1,\dots,Q_{t^\ast}=T]$2 is roughly $\Pi=[Q_0=S,Q_1,\dots,Q_{t^\ast}=T]$3 times or less than the simple method’s across the tested maps (Suzuki et al., 22 Oct 2025).

The application domains named for CUMAPF and closely related connected planning include self-reconfiguration, marching, platooning, swarm navigation, programmable matter, and search and rescue (Suzuki et al., 22 Oct 2025, Bui et al., 6 Jan 2025, Queffelec et al., 2020). In these settings, standard MAPF is insufficient precisely because it does not guarantee preservation of team connectivity (Suzuki et al., 22 Oct 2025).

Several limitations are explicit in the supplied material. PULL is complete and polynomial-time, but not globally optimal (Suzuki et al., 22 Oct 2025). LaCAM* combined with PULL is eventually optimal, but difficult instances may remain expensive under substantial time budgets (Suzuki et al., 22 Oct 2025). TP-SWAP is complete for decentralized AMAPF under its communication and synchronization assumptions, but does not impose a single always-connected communication network (Dergachev et al., 2024). MALCR’s MA-DL framework is incomplete because its greedy SAPF module returns the first shortest valid path it finds rather than exploring all alternatives, and the formulation is labeled rather than unlabeled (Bui et al., 6 Jan 2025). CCBS provides an optimal exact algorithm only in its Neg-only variant; its more aggressive positive-constraint branching is incomplete in general, and its core CMAPF formulation does not include the standard MAPF collision model (Queffelec et al., 2020).

Taken together, these results place CUMAPF as a distinct research problem rather than a minor extension of existing MAPF variants. Its defining feature is the requirement that an interchangeable multi-agent team move without fragmenting, and the main current algorithmic directions are rule-based connected configuration generation, exact search with connectivity-aware successor generation, and decentralized unlabeled coordination under local communication assumptions (Suzuki et al., 22 Oct 2025, Dergachev et al., 2024).