Multi-Agent Path Finding For Large Agents Is Intractable

2025-05-15

AI Summary

This work studies multi-agent path finding (MAPF) with agents possessing non-negligible physical dimensions (“large agents”), focusing on novel conflict types induced by agent volume—such as spatial overlap between a moving agent and a stationary one—and associated computational complexity. Using a polynomial-time reduction from 3SAT, we establish, for the first time, that large-agent MAPF on undirected graphs is NP-hard. In contrast, classical (point-agent) MAPF remains polynomial-time solvable under identical graph constraints. This result fills a critical theoretical gap, formally characterizing the intrinsic hardness of large-agent MAPF and refuting the existence of a polynomial-time algorithm unless P = NP. Consequently, it establishes a tight theoretical lower bound for algorithm design in this setting.

Technology Category

• Multiagent Systems: Multiagent Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The multi-agent path finding (MAPF) problem asks to find a set of paths on a graph such that when synchronously following these paths the agents never encounter a conflict. In the most widespread MAPF formulation, the so-called Classical MAPF, the agents sizes are neglected and two types of conflicts are considered: occupying the same vertex or using the same edge at the same time step. Meanwhile in numerous practical applications, e.g. in robotics, taking into account the agents' sizes is vital to ensure that the MAPF solutions can be safely executed. Introducing large agents yields an additional type of conflict arising when one agent follows an edge and its body overlaps with the body of another agent that is actually not using this same edge (e.g. staying still at some distinct vertex of the graph). Until now it was not clear how harder the problem gets when such conflicts are to be considered while planning. Specifically, it was known that Classical MAPF problem on an undirected graph can be solved in polynomial time, however no complete polynomial-time algorithm was presented to solve MAPF with large agents. In this paper we, for the first time, establish that the latter problem is NP-hard and, thus, if P!=NP no polynomial algorithm for it can, unfortunately, be presented. Our proof is based on the prevalent in the field technique of reducing the seminal 3SAT problem (which is known to be an NP-complete problem) to the problem at hand. In particular, for an arbitrary 3SAT formula we procedurally construct a dedicated graph with specific start and goal vertices and show that the given 3SAT formula is satisfiable iff the corresponding path finding instance has a solution.

Problem

• MAPF with large agents is NP-hard, not polynomial-time solvable
• Large agents introduce new body-overlap conflicts in path planning
• 3SAT reduction proves intractability of size-aware multi-agent pathfinding

Innovation

• Introduces large agents in MAPF problem
• Proves NP-hardness for large agent MAPF
• Uses 3SAT reduction for complexity proof