Many-to-Many Multi-Agent Pickup and Delivery

• Many-to-Many Multi-Agent Pickup and Delivery is a dynamic framework where robots serve multiple tasks with interchangeable pickup and delivery points in a shared workspace.
• Algorithms decouple assignment and collision-free path planning, using strategies like LNS, PBS, and flow-based methods to efficiently address high-dimensional decision spaces.
• Empirical studies show that integrated assignment methods and congestion-aware routing significantly improve throughput and service time in dense, real-time warehouse operations.

Searching arXiv for recent and foundational papers on many-to-many multi-agent pickup and delivery, MAPD, and closely related formulations. Many-to-Many Multi-Agent Pickup and Delivery (M2M) denotes a class of lifelong multi-robot transportation problems in which multiple agents repeatedly serve multiple pickup-and-delivery requests while sharing a common workspace and avoiding collisions. In its most specific warehouse interpretation, a task is not tied to a unique pickup and delivery endpoint; rather, it may admit multiple feasible source and destination locations, turning assignment into a coupled choice over agents, tasks, pickup locations, and delivery locations (Schneider et al., 8 May 2026). More broadly, the term is also used for repeated many-agent/many-task pickup-and-delivery systems in which requests arrive over time, agents must be routed through shared infrastructure, and execution quality depends jointly on task allocation, route sequencing, and multi-agent motion coordination (Ma et al., 2017, Xu et al., 2022). The literature therefore contains both a narrow, explicit many-to-many formulation with interchangeable endpoints (Schneider et al., 8 May 2026) and a wider family of MAPD models that capture important M2M substructures such as online assignment, multi-goal routing, capacity, transfers, and deadlock-free execution (Chen et al., 2021, Zhang et al., 7 Aug 2025).

1. Formal scope and problem variants

The canonical MAPD substrate models the environment as an undirected graph, with agents moving synchronously in discrete time and obeying standard MAPF collision constraints: no two agents may occupy the same vertex at the same time, and no two agents may swap along the same edge in opposite directions simultaneously (Xu et al., 2022, Ma et al., 2017). In this baseline setting, each task consists of a release time, one pickup location, and one delivery location; an assigned agent must first reach the pickup, then the delivery, and becomes free only after completing the task (Ma et al., 2017, Shah et al., 21 Apr 2025). This formulation already yields a many-agents/many-tasks system over time, but each individual request remains one-to-one.

The explicit M2M formulation generalizes the task itself. In "Many-to-Many Multi-Agent Pickup and Delivery" (Schneider et al., 8 May 2026), each task is defined as $\tau_n=(S_n,D_n)$, where $S_n \subseteq V$ is the set of admissible pickup locations and $D_n \subseteq V$ is the set of admissible delivery locations. Once assigned to agent $a_m$, the instantiated task becomes $\tau_n^m=(s_p,d_q)$ with $s_p\in S_n$ and $d_q\in D_n$. The resulting decision space is four-dimensional over agents, tasks, source choices, and destination choices; the paper characterizes this as a four-dimensional assignment problem and notes the combinatorial scale $M\times N\times P\times Q$ in a naive representation (Schneider et al., 8 May 2026).

A second line of generalization replaces the fixed pickup-delivery pair by an ordered sequence of goals. In Multi-Goal MAPD (MG-MAPD), each task has a release time and a sequence of goal locations that must be visited in order; standard MAPD is the special case of length two (Xu et al., 2022). This permits structured missions such as one pickup followed by several deliveries, but the internal order remains fixed and the framework still assumes that, once an agent starts a task, it cannot execute other tasks simultaneously (Xu et al., 2022).

A third axis of variation introduces capacity and route interleaving. "Integrated Task Assignment and Path Planning for Capacitated Multi-Agent Pickup and Delivery" (Chen et al., 2021) allows each robot to carry multiple tasks simultaneously up to capacity $C$, with load evolution and pickup-before-delivery precedence enforced along an ordered action sequence. This moves beyond standard one-task-at-a-time MAPD and is materially closer to many-to-many routing, although each request remains a single pickup-delivery pair assigned to one robot (Chen et al., 2021).

These variants delimit the current meaning of M2M in the literature. In the strictest sense, M2M means task definitions with multiple admissible pickups and deliveries (Schneider et al., 8 May 2026). In a broader robotics sense, it also encompasses repeated many-agent/many-task pickup-and-delivery with online arrivals, congestion, and route-level sequencing, even when each atomic request still has one pickup and one drop-off (Zhang et al., 7 Aug 2025, Shah et al., 21 Apr 2025).

2. Mathematical structure and objectives

At the motion layer, most M2M/MAPD papers share the same feasibility model. On a graph $G=(V,E)$, each agent location $S_n \subseteq V$0 or $S_n \subseteq V$1 must evolve by waiting or traversing an adjacent edge; vertex conflicts and head-on edge conflicts are forbidden (Xu et al., 2022, Schneider et al., 8 May 2026, Ma et al., 2017). This common substrate enables the reuse of MAPF solvers such as PBS, ECBS, CBS, or prioritized space-time search as low-level collision-resolution mechanisms.

What differs is the optimization objective. Classical one-shot MAPF objectives such as makespan or sum of costs are generally not primary in lifelong pickup-and-delivery. The dominant effectiveness metric is service time or a closely related delay measure. In MG-MAPD, the service time of task $S_n \subseteq V$2 is

$S_n \subseteq V$3

and the empirical aim is to minimize average service time over all tasks (Xu et al., 2022). The earlier lifelong MAPD formulation likewise evaluates algorithms by average service time and defines solvability as bounded service time for all tasks (Ma et al., 2017). Neural ATTF also optimizes average service time, reporting throughput and runtime as complementary metrics (Shah et al., 21 Apr 2025).

The explicit M2M warehouse formulation instead states a system-level objective of maximizing throughput in tasks/minute under an online planning budget such as one second, while operationalizing the optimization through surrogate assignment costs based on estimated task durations (Schneider et al., 8 May 2026). Its base assignment cost is

$S_n \subseteq V$4

subject to validity and queue-capacity constraints, and the final quality criterion is empirical throughput over long simulation horizons (Schneider et al., 8 May 2026). The SKU-aware variant augments this with nearest-neighbor SKU-distribution terms, although in reported experiments the throughput gains of that augmentation were limited relative to pure travel-time minimization (Schneider et al., 8 May 2026).

Flow-based online MAPD takes yet another perspective by formulating the assignment subproblem directly as a minimum-cost flow on the environment graph (Zhang et al., 7 Aug 2025). There, the objective is the min-cost flow objective

$S_n \subseteq V$5

with unit, planner-based, or historical congestion-aware edge costs. With unit edge costs, the formulation yields globally optimal assignment with respect to shortest-path distance from free agents to pickup nodes, without explicitly materializing all pairwise agent-task distances (Zhang et al., 7 Aug 2025). This is an assignment-layer objective rather than an end-to-end pickup-delivery social cost, but it addresses the main real-time bottleneck in large online systems.

Capacity-aware formulations use richer objectives. In capacitated MAPD, the total travel delay objective is

$S_n \subseteq V$6

measuring excess completion time beyond release plus direct shortest pickup-to-delivery travel (Chen et al., 2021). In dynamic perturbation-aware multi-commodity PDVRP, the objective is total fleet travel distance under payload and precedence constraints, but without collision-aware motion (Goutham et al., 2023). These differences matter because they distinguish routing-centric M2M work from lifelong service-centric MAPD work.

3. Core algorithmic decompositions

A dominant architectural pattern is decoupling with controlled coupling: task assignment is solved at one layer, collision-free path planning at another, but information is exchanged through estimated costs, reservations, guide paths, or replanning triggers.

LNS-PBS and LNS-wPBS exemplify this pattern at the route-sequencing level (Xu et al., 2022). Each agent maintains a dummy endpoint, a task sequence, a concatenated goal sequence, and a collision-free path through that sequence. Large Neighborhood Search assigns or repairs multi-task sequences using estimated service time under shortest-path assumptions, while Priority-Based Search or windowed PBS computes collision-free paths through the resulting goal sequences (Xu et al., 2022). Because the assignment layer reasons over short future sequences rather than only the next task, the method supports limited route-level optimization even in the lifelong setting.

The explicit M2M algorithm of (Schneider et al., 8 May 2026) uses a different decomposition. It starts from a conceptual 4D tensor

$S_n \subseteq V$7

with entries $S_n \subseteq V$8 for assigning agent $S_n \subseteq V$9 to task $D_n \subseteq V$0 via source $D_n \subseteq V$1 and destination $D_n \subseteq V$2. To avoid constructing this tensor directly, it decomposes the structure into four lower-dimensional matrices: $D_n \subseteq V$3 A greedy initial assignment is then improved by LNS with Shaw removal and simulated-annealing-style acceptance, after which PBS plans collision-free motion segment by segment (Schneider et al., 8 May 2026). The key novelty is that source and destination selection are optimized jointly with agent-task assignment rather than being preprocessed away.

Flow-based task assignment changes the decomposition more radically (Zhang et al., 7 Aug 2025). Instead of building a bipartite graph over agent-task pairs, it embeds the assignment directly on the map: free-agent locations connect to a dummy source, pickup nodes connect to a dummy sink, and map edges carry traversal cost. Solving min-cost flow simultaneously determines which free agent takes which task and yields a route skeleton to the pickup. The extracted guide paths are then passed to Guided PIBT for collision-free execution (Zhang et al., 7 Aug 2025). This directly addresses extreme-scale settings where pairwise distance computation dominates runtime.

A further variant combines lightweight task matching with learned space-time planning. Neural ATTF uses Priority Guided Task Matching to pair delayed or idle agents with currently unassigned tasks, then Neural STA* plans collision-free trajectories in a time-augmented state space using learned guidance maps (Shah et al., 21 Apr 2025). This keeps assignment simple but compensates through a more scalable planner, yielding favorable runtime-quality tradeoffs in dense online settings (Shah et al., 21 Apr 2025).

These decompositions reveal a consistent theme: M2M difficulty is rarely concentrated in a single subproblem. Endpoint choice, route sequencing, congestion-aware dispatch, and low-level collision management interact strongly, and the most successful methods expose these interactions selectively rather than optimizing everything monolithically.

4. Execution models and coordination regimes

The literature spans centralized, token-based, distributed, and learning-based coordination regimes.

The classical lifelong MAPD regime is token-based. Token Passing and Token Passing with Task Swaps maintain a synchronized shared token containing current paths, assignments, and unexecuted tasks (Ma et al., 2017). Agents request the token when their current path ends, assign themselves a feasible task or move to a parking endpoint, and plan collision-free paths against existing token reservations (Ma et al., 2017). TPTS extends TP by allowing reassignment before pickup via recursive task swaps, improving service time at higher computational cost (Ma et al., 2017).

Dynamic order-evolution work preserves this token-passing structure while making task content mutable. In Dynamic-MAPD for robotic cellular warehousing, an order may gain new SKU pickups during execution; Dynamic Token Passing replans only the affected order when such an event occurs, and Cooperative Token Passing may delegate newly added pickups to idle robots if doing so reduces estimated completion time (Ren et al., 4 Jun 2026). This introduces partial order sharing across robots, but still in a highly structured many-pickups-to-one-delivery setting (Ren et al., 4 Jun 2026).

Distributed coordination regimes instead rely on local reservation and traffic structure. The asynchronous MAPD method of (Miyashita et al., 2023) organizes the environment into a strongly oriented main area plus attached trees, uses node agents to manage next-node reservations, and permits local wait/detour responses under fluctuating travel times. Its contribution is not richer assignment but robust execution under delays, non-WFI endpoints, and limited endpoint availability (Miyashita et al., 2023). PIBTTP and PIBTTP-TA similarly focus on deadlock-free local motion coordination in graphs with tree-shaped dead-end subgraphs attached to a bi-connected main area; temporary priorities help agents leave trees without deadlock (Fujitani et al., 2022).

Learning-based coordination offers yet another execution regime. SePar treats multi-agent path planning as autoregressive sequence prediction over agents and argues that this yields implicit information exchange and order-invariant optimality of the induced policy space (Zhao et al., 28 Sep 2025). Its practical domain is warehouse MAPD-like execution rather than assignment, and its direct relevance to M2M lies in coordination under extreme corridor congestion rather than in task-level modeling (Zhao et al., 28 Sep 2025). Neural ATTF likewise uses a centralized planner at each timestep but relies on learned low-level guidance rather than exact multi-agent search (Shah et al., 21 Apr 2025).

Transfer-based cooperation occupies a separate niche. DELIVER studies a single pickup-delivery mission executed through a chain of relay robots chosen by Voronoi-region intersection with a global path; relay points are selected on shared Voronoi boundaries using a minimax distance criterion (Srivastava et al., 26 Aug 2025). This is not many-to-many in the standard sense, but it is relevant because it introduces explicit inter-robot handoff, a feature largely absent from mainstream MAPD formulations (Srivastava et al., 26 Aug 2025).

5. Structural assumptions, completeness, and deadlock avoidance

A conspicuous feature of M2M/MAPD research is the reliance on structural assumptions that make lifelong execution provably feasible.

The most influential is well-formedness. In lifelong MAPD, a well-formed instance requires finitely many tasks, at least as many non-task endpoints as agents, and endpoint-to-endpoint paths that avoid other endpoints (Ma et al., 2017). Under these assumptions, TP and TPTS solve all such instances (Ma et al., 2017). MG-MAPD inherits a related notion: start locations must be distinct from task endpoints, and any two endpoints must be connected by a path traversing no other endpoints (Xu et al., 2022). This endpoint separability underpins the completeness proof for LNS-PBS, whose modified PBS uses old paths as reservations and dummy endpoints as private waiting locations (Xu et al., 2022).

These completeness guarantees are valuable but narrow. They ensure eventual completion of all finite tasks on well-formed instances, not service-time optimality, bounded suboptimality, or robustness under arbitrary maps. Windowed variants such as LNS-wPBS sacrifice completeness for scalability and runtime stability (Xu et al., 2022).

In less structured environments, deadlock avoidance often substitutes for full completeness proofs. SBDA addresses maze-like graphs with dead-end endpoints by dynamically identifying standby nodes that are not articulation points, not endpoints, and not dead ends; reserving such nodes preserves graph connectivity for the remainder of the system (Yamauchi et al., 2022). By coupling standby selection with task admissibility conditions around congested endpoints, SBDA solves well-formed MAPD instances in environments where endpoint holding alone performs poorly (Yamauchi et al., 2022).

Similarly, PIBTTP’s correctness rests on graph structure: a bi-connected main area plus attached trees, together with restrictions such as not assigning pickup and delivery in the same tree (Fujitani et al., 2022). Dynamic asynchronous execution methods impose their own graph conditions, such as decomposition into a connected main area and attached trees with sufficient slack nodes (Miyashita et al., 2023). These assumptions indicate that motion-layer guarantees remain highly topology-dependent.

A different type of guarantee appears in continuous-time planning. TP-SIPPwRT replaces discrete token-passing path planning with Safe Interval Path Planning with Reservation Table, incorporating non-holonomic orientation, translational and rotational velocities, and safety radii directly in planning. The method returns collision-free paths and yields a TP-based algorithm complete for well-formed MAPD instances while planning directly in continuous time (Ma et al., 2018). For M2M, this is chiefly relevant as a low-level execution primitive rather than a task model.

6. Scalability and empirical regimes

The empirical literature supports several distinct claims about M2M/MAPD performance.

At moderate scale on classical online MAPD benchmarks, route-sequencing plus PBS improves service quality. LNS-PBS is reported to achieve up to a $D_n \subseteq V$4 service-time advantage over CENTRAL on small online/offline MAPD, while LNS-wPBS is more efficient and more stable and scales to large warehouses with thousands of agents and tasks (Xu et al., 2022). On medium and large settings, LNS-wPBS improves service time over HBH+MLA* by more than $D_n \subseteq V$5 and up to $D_n \subseteq V$6 in medium instances, and by about $D_n \subseteq V$7 to $D_n \subseteq V$8 with 1000 agents and up to 5000 tasks in large instances, though with higher runtime per timestep (Xu et al., 2022).

At very large scale, the dominant bottleneck shifts to assignment. Flow-based task assignment scales to over 20000 agents and 30000 tasks within a 1-second planning budget, because complexity depends primarily on map size rather than the full $D_n \subseteq V$9 agent-task pair set (Zhang et al., 7 Aug 2025). In congested settings, congestion-aware flow costs outperform both greedy matching and unit-cost flow. For example, Flow-Traffic improves throughput by $a_m$0 over Greedy on Warehouse Small with 600 agents and by $a_m$1 on Sortation Large with 20000 agents under strict real-time constraints (Zhang et al., 7 Aug 2025).

The explicit many-to-many warehouse formulation (Schneider et al., 8 May 2026) shows the clearest evidence that endpoint choice matters as a first-class decision. With 40 agents over 8-hour simulations, M2M can complete up to 22,132 additional tasks on average relative to LNS-PBS after converting many-to-many tasks to one-to-one, with the largest gains appearing in constrained layouts and high inventory density (Schneider et al., 8 May 2026). On the restricted map at 90% density, throughput improves by $a_m$2 over the converted one-to-one baseline (Schneider et al., 8 May 2026). Runtime remains close to the baseline, around 1.15 seconds versus 1.07 seconds on average for task allocation (Schneider et al., 8 May 2026).

Learning-based methods display a different regime. Neural ATTF reports strong service-time/runtime tradeoffs and favorable scaling to hundreds of agents on warehouse maps, with runtime remaining far below heavyweight centralized baselines at high demand frequencies (Shah et al., 21 Apr 2025). SePar shows that learning-based pathfinding can retain 50%–90% of LaCAM2’s performance beyond 128 agents in warehouse settings where other learning baselines collapse, although its contribution remains confined to routing rather than assignment (Zhao et al., 28 Sep 2025). By contrast, the earlier MAPF-vs-MARL comparison found that CBS outperformed SEAC at small scale but scaled poorly beyond about 10 agents, while SEAC retained stable execution-time responsiveness at the cost of worse path efficiency (Lau et al., 2022).

Capacity and route interleaving also materially change performance. In capacitated MAPD, increasing robot capacity from 1 to 3 or 5 dramatically reduces delay and makespan in lifelong settings; for 50 agents at task frequency 10, total travel delay drops from 38050 at capacity 1 to 15851 at capacity 3 and 11464 at capacity 5 (Chen et al., 2021). This suggests that many-to-many benefits may depend as much on transport semantics as on assignment sophistication.

7. Relation to adjacent pickup-and-delivery research

M2M MAPD sits between multi-agent pathfinding and vehicle routing, borrowing machinery from both but matching neither exactly.

Relative to MAPF, M2M inherits collision-free multi-agent motion on shared maps but replaces one-shot fixed goals with an online stream of pickup-and-delivery goals and service-oriented objectives (Ma et al., 2017, Xu et al., 2022). Relative to classical PDP/VRP, it inherits multi-request assignment, sequencing, and capacity structure, but introduces shared-space collision constraints and often much finer-grained motion models (Chen et al., 2021, Ma et al., 2018). The resulting hybrid character explains why decoupled assignment-plus-MAPF pipelines remain common.

Vehicle-routing papers become most relevant when capacity, multi-commodity structure, or perturbation recovery are central. The m-PDVRP work on recomputing perturbed multi-commodity pickup-and-delivery solutions with Monte Carlo Tree Search addresses unique commodities, heterogeneous capacities, and search-tree reuse under perturbations, but omits collision-aware path planning and online arrivals (Goutham et al., 2023). The multi-purpose PDP for combined passenger and freight transport introduces platform-module decomposition and en-route capability switching via service depots, but is a centralized static fleet-routing model rather than a multi-agent coordination model (Hatzenbühler et al., 2022). These works inform the transport layer of M2M without addressing the MAPF layer.

Transfer-enabled cooperation remains underexplored in mainstream M2M MAPD. DELIVER demonstrates handoff-based single-mission execution (Srivastava et al., 26 Aug 2025), and Cooperative Token Passing permits limited delegation of newly appended pickups within a dynamic order (Ren et al., 4 Jun 2026), but general transshipment among agents is still outside the standard formal scope. Likewise, time windows, synchronization constraints, split deliveries, and dynamic many-to-many matching across source and sink sets remain largely absent from robotics MAPD formulations (Xu et al., 2022, Schneider et al., 8 May 2026).

This suggests a useful taxonomy. One branch studies routing-and-motion M2M, emphasizing collision-free lifelong execution under shared-space constraints (Xu et al., 2022, Zhang et al., 7 Aug 2025, Shah et al., 21 Apr 2025). Another studies transport-structure M2M, emphasizing capacities, commodities, reconfiguration, or perturbations without shared-space collision avoidance (Chen et al., 2021, Goutham et al., 2023, Hatzenbühler et al., 2022). The explicit many-to-many warehouse model (Schneider et al., 8 May 2026) is notable because it bridges these branches by introducing interchangeable sources and destinations while retaining multi-agent collision-free execution.

8. Conceptual status and open directions

The current literature supports a clear but nuanced interpretation of M2M. The term does not yet denote a single unified canonical problem. Instead, it spans a hierarchy of formulations.

At the narrow end, M2M means tasks with multiple admissible source and destination sets, solved online with joint choice of agent, pickup location, and delivery location under collision-aware execution (Schneider et al., 8 May 2026). At an intermediate level, it includes route-level sequencing over multiple future tasks or multiple goals per task, as in MG-MAPD and capacitated MAPD (Xu et al., 2022, Chen et al., 2021). At the broadest robotics level, it also encompasses lifelong many-agent/many-task systems whose essential difficulty lies in shared-space routing, congestion, and continual replanning, even if each atomic task remains a single pickup-delivery pair (Zhang et al., 7 Aug 2025, Shah et al., 21 Apr 2025).

Several open directions follow directly from the surveyed work. One is integrated many-to-many assignment under congestion, combining interchangeable endpoints with route sequencing and collision-aware cost estimation at scales beyond the current 40-agent M2M experiments (Schneider et al., 8 May 2026). Another is capacity and load-state integration in lifelong shared-space systems, extending current unit-demand or small-capacity models toward full multi-commodity transport (Chen et al., 2021, Goutham et al., 2023). A third is transfer-enabled M2M, where explicit handoff and transshipment are optimized rather than handled as special cases (Srivastava et al., 26 Aug 2025, Ren et al., 4 Jun 2026). A fourth is robust dynamic execution, including asynchronous travel times, perturbations, and dynamic order evolution beyond one update or one active mission structure (Miyashita et al., 2023, Ren et al., 4 Jun 2026, Goutham et al., 2023). A plausible implication is that future M2M systems will remain hierarchical, with high-level transport optimization, mid-level assignment or route-fragment reasoning, and low-level collision-aware execution coupled by learned or analytical surrogates rather than solved monolithically.

The field’s main misconception is to equate lifelong MAPD with full many-to-many pickup-and-delivery. The literature shows they overlap but are not identical. Standard MAPD solves repeated one-pickup/one-delivery execution with collision avoidance (Ma et al., 2017). M2M, in its stronger sense, requires additional choice over admissible sources and sinks, route interleaving, load state, or commodity coupling (Schneider et al., 8 May 2026, Chen et al., 2021). Conversely, operations-research M2M models often omit the shared-space collision dynamics that dominate real multi-robot warehouses (Hatzenbühler et al., 2022, Goutham et al., 2023). The technical challenge of M2M is precisely this jointness: transportation flexibility and multi-agent motion are both first-order, and neither can be cleanly abstracted away without losing critical problem structure.