In the Multiagent Path Finding problem (MAPF for short), we focus on efficiently finding non-colliding paths for a set of $k$ agents on a given graph $G$, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule $\ell$, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our fixed-parameter tractability results. We show that MAPF is W[1]-hard with respect to $k$ (even if $k$ is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan$\ell$ are fixed constants. On the positive side, we show an FPT algorithm for $k+\ell$. As we delve further, the structure of~$G$ comes into play. We give an FPT algorithm for parameter $k$ plus the diameter of the graph~$G$. The MAPF problem is W[1]-hard for cliquewidth of $G$ plus $\ell$ while it is FPT for treewidth of $G$ plus $\ell$.

翻译：在多智能体路径规划问题（简称MAPF）中，我们专注于在一个给定图$G$上为一组$k$个智能体高效地找到无碰撞路径，其中每个智能体需从起点顶点前往目标。解质量的一个重要衡量标准是所提出调度$\ell$的长度，即最长路径（包含等待时间）的长度。在本工作中，我们在参数化复杂度框架下开展系统性研究。我们提供的困难性结果与该问题常用的多种启发式算法相吻合，而这些算法的运行时间可能基于我们的固定参数可解性结果而得到改进。我们证明MAPF关于参数$k$是W[1]-难的（即使将$k$与输入图的最大度结合考虑）。在平面图中该问题仍为NP-难，即使最大度与完工时间$\ell$均为固定常数。从积极方面看，我们针对$k+\ell$给出了一个FPT算法。进一步研究时，图$G$的结构开始发挥作用：我们针对参数$k$结合图$G$的直径给出了一个FPT算法。MAPF问题对于图$G$的团宽加上$\ell$是W[1]-难的，而对于图$G$的树宽加上$\ell$则是FPT的。