We study a fundamental NP-hard motion coordination problem for multi-robot/multi-agent systems: We are given a graph $G$ and set of agents, where each agent has a given directed path in $G$. Each agent is initially located on the first vertex of its path. At each time step an agent can move to the next vertex on its path, provided that the vertex is not occupied by another agent. The goal is to find a sequence of such moves along the given paths so that each reaches its target, or to report that no such sequence exists. The problem models guidepath-based transport systems, which is a pertinent abstraction for traffic in a variety of contemporary applications, ranging from train networks or Automated Guided Vehicles (AGVs) in factories, through computer game animations, to qubit transport in quantum computing. It also arises as a sub-problem in the more general multi-robot motion-planning problem. We provide a fine-grained tractability analysis of the problem by considering new assumptions and identifying minimal values of key parameters for which the problem remains NP-hard. Our analysis identifies a critical parameter called vertex multiplicity (VM), defined as the maximum number of paths passing through the same vertex. We show that a prevalent variant of the problem, which is equivalent to Sequential Resource Allocation (concerning deadlock prevention for concurrent processes), is NP-hard even when VM is 3. On the positive side, for VM $\le$ 2 we give an efficient algorithm that iteratively resolves cycles of blocking relations among agents. We also present a variant that is NP-hard when the VM is 2 even when $G$ is a 2D grid and each path lies in a single grid row or column. By studying highly distilled yet NP-hard variants, we deepen the understanding of what makes the problem intractable and thereby guide the search for efficient solutions under practical assumptions.

翻译：我们研究多机器人/多智能体系统中一个基础的NP难运动协调问题：给定图$G$和一组智能体，每个智能体在$G$中具有一条有向路径。每个智能体初始位于其路径的第一个顶点上。在每个时间步，智能体可沿路径移动至下一顶点，但前提是该顶点未被其他智能体占用。目标是找到一组沿指定路径的移动序列，使得每个智能体到达其目标点，或判定不存在这样的序列。该问题建模了基于导引路径的运输系统，这是多种当代应用场景中交通问题的相关抽象，涵盖铁路网络、工厂中的自动导引车（AGVs）、计算机游戏动画，乃至量子计算中的量子比特传输。它也是更一般的多机器人运动规划问题的子问题。我们通过考虑新假设并识别问题保持NP难的关键参数最小值，对该问题进行了细粒度的可解性分析。分析揭示了一个关键参数——顶点多重度（VM），定义为通过同一顶点的最大路径数。我们证明，该问题的一个常见变体（等价于顺序资源分配，涉及并发进程的死锁预防）即使在VM为3时仍是NP难的。从积极方面看，对于VM ≤ 2，我们提出了一种高效算法，该算法迭代地解决智能体间阻塞关系的循环。我们还提出一个变体，当VM为2时，即使G是二维网格且每条路径位于单行或单列中，该问题仍是NP难的。通过研究高度精炼但保持NP难的变体，我们加深了对问题难解性的理解，从而指导在实用假设下寻求高效解法的方向。