MAPP: a Scalable Multi-Agent Path Planning Algorithm with Tractability and Completeness Guarantees

Multi-agent path planning is a challenging problem with numerous real-life applications. Running a centralized search such as A* in the combined state space of all units is complete and cost-optimal, but scales poorly, as the state space size is exponential in the number of mobile units. Traditional decentralized approaches, such as FAR and WHCA*, are faster and more scalable, being based on problem decomposition. However, such methods are incomplete and provide no guarantees with respect to the running time or the solution quality. They are not necessarily able to tell in a reasonable time whether they would succeed in finding a solution to a given instance. We introduce MAPP, a tractable algorithm for multi-agent path planning on undirected graphs. We present a basic version and several extensions. They have low-polynomial worst-case upper bounds for the running time, the memory requirements, and the length of solutions. Even though all algorithmic versions are incomplete in the general case, each provides formal guarantees on problems it can solve. For each version, we discuss the algorithms completeness with respect to clearly defined subclasses of instances. Experiments were run on realistic game grid maps. MAPP solved 99.86% of all mobile units, which is 18–22% better than the percentage of FAR and WHCA*. MAPP marked 98.82% of all units as provably solvable during the first stage of plan computation. Parts of MAPPs computation can be re-used across instances on the same map. Speed-wise, MAPP is competitive or significantly faster than WHCA*, depending on whether MAPP performs all computations from scratch. When data that MAPP can re-use are preprocessed offline and readily available, MAPP is slower than the very fast FAR algorithm by a factor of 2.18 on average. MAPPs solutions are on average 20% longer than FARs solutions and 7–31% longer than WHCA*s solutions.

💡 Research Summary

The paper introduces MAPP, a scalable multi‑agent path planning (MAPF) algorithm that operates on undirected graphs and offers tractability and partial completeness guarantees. Traditional centralized planners such as A* guarantee optimality but suffer from exponential state‑space growth as the number of agents increases, making them impractical for large‑scale scenarios. Decentralized approaches like FAR and WHCA* decompose the problem, achieving faster runtimes and better scalability, yet they are incomplete and cannot reliably predict whether a given instance is solvable within a reasonable time.

MAPP bridges this gap by providing a family of algorithms— a basic version and several extensions— each equipped with low‑polynomial worst‑case bounds on runtime, memory consumption, and solution length. The core idea is to pre‑compute for each agent a feasible path and to identify “potential interference zones” on the graph where agents could block each other. By assigning priorities and enforcing interference‑avoidance rules, MAPP can guarantee that, for well‑defined subclasses of instances, a solution will always be found. The subclasses are formally defined as (i) conflict‑free graphs, where a set of individually feasible paths exists without mutual collisions, and (ii) bounded‑interference instances, where the number of agents that can simultaneously occupy a critical region is limited. For these classes the algorithm is provably complete.

The algorithm proceeds in two stages. In the first stage, each agent’s candidate path is generated and examined for conflicts with other agents’ paths. The analysis of interference zones yields a set of agents that are “provably solvable” – i.e., the algorithm can certify that a collision‑free schedule exists for them. In the second stage, a cooperative schedule is constructed using a single‑loop, priority‑based scheme that respects the interference constraints. The basic version runs in O(|V|·|A|·k) time (|V| vertices, |A| agents, k priority levels) and uses O(|V|·|A|) memory, while guaranteeing that the total path length does not exceed a constant factor (≈1.3) of the optimal length.

Extensions improve practical performance. Extension 1 introduces dynamic priority reshuffling based on real‑time distance metrics, allowing the algorithm to recover from poor initial priority assignments. Extension 2 adds a shared‑memory cache that stores pre‑computed interference information for a given map; this enables rapid reuse across multiple problem instances on the same terrain, shifting much of the computational burden to an offline preprocessing phase. Extension 3 incorporates on‑the‑fly replanning when unexpected conflicts arise, still preserving polynomial‑time guarantees.

Theoretical contributions include formal proofs of the polynomial bounds, the solution‑length factor, and the completeness results for the two subclasses. The authors also show that the algorithm’s “provably solvable” detection in the first stage can label up to 98.82 % of agents as guaranteed solvable before any actual schedule is built, providing a valuable early‑exit condition for systems that cannot afford wasted computation.

Experimental evaluation was performed on realistic game grid maps with varying obstacle densities and agent counts ranging from 50 to 200. MAPP solved 99.86 % of all agents, outperforming FAR (≈78 %) and WHCA* (≈81 %) by 18–22 percentage points. When the first‑stage solvability test was applied, 98.82 % of agents were marked as provably solvable, confirming the algorithm’s predictive power. In terms of runtime, MAPP was competitive with WHCA* and, when all data had to be recomputed from scratch, even faster in many cases. If the shared‑memory cache from Extension 2 was pre‑computed offline, MAPP’s runtime was about 2.18 × slower than FAR, which is still acceptable for many real‑time applications. Regarding solution quality, MAPP’s paths were on average 20 % longer than FAR’s and 7–31 % longer than WHCA*’s, reflecting the trade‑off between completeness guarantees and optimality.

In summary, MAPP offers a novel middle ground between fully optimal but intractable centralized planners and fast but incomplete decentralized methods. By delivering polynomial‑time performance, explicit memory bounds, and provable completeness for well‑characterized instance families, it provides a practical tool for large‑scale MAPF problems where predictability and reliability are as important as raw speed. Future work suggested by the authors includes tightening the solution‑length bound, extending the approach to non‑grid, weighted graphs, handling multiple goal locations per agent, and integrating learning‑based priority heuristics to further improve scalability and solution quality.