Multi-Agent Path Finding (MAPF) is NP-hard to solve optimally, even on graphs, suggesting no polynomial-time algorithms can compute exact optimal solutions for them. This raises a natural question: How optimal can polynomial-time algorithms reach? Whereas algorithms for computing constant-factor optimal solutions have been developed, the constant factor is generally very large, limiting their application potential. In this work, among other breakthroughs, we propose the first low-polynomial-time MAPF algorithms delivering $1$-$1.5$ (resp., $1$-$1.67$) asymptotic makespan optimality guarantees for 2D (resp., 3D) grids for random instances at a very high $1/3$ agent density, with high probability. Moreover, when regularly distributed obstacles are introduced, our methods experience no performance degradation. These methods generalize to support $100\%$ agent density. Regardless of the dimensionality and density, our high-quality methods are enabled by a unique hierarchical integration of two key building blocks. At the higher level, we apply the labeled Grid Rearrangement Algorithm (RTA), capable of performing efficient reconfiguration on grids through row/column shuffles. At the lower level, we devise novel methods that efficiently simulate row/column shuffles returned by RTA. Our implementations of RTA-based algorithms are highly effective in extensive numerical evaluations, demonstrating excellent scalability compared to other SOTA methods. For example, in 3D settings, \rta-based algorithms readily scale to grids with over $370,000$ vertices and over $120,000$ agents and consistently achieve conservative makespan optimality approaching $1.5$, as predicted by our theoretical analysis.

翻译：多智能体路径规划（MAPF）问题即使在图结构上也是NP难问题，这意味着不存在多项式时间算法能够为其计算精确最优解。这引发了一个自然问题：多项式时间算法能够达到何种最优性？尽管已有算法能够计算常数倍最优解，但该常数通常非常大，限制了其应用潜力。在本工作中，我们取得了多项突破，其中包括提出了首个低多项式时间MAPF算法，该算法以高概率为随机实例（在高达$1/3$的智能体密度下）在二维（对应地，三维）网格上提供$1$-$1.5$（对应地，$1$-$1.67$）的渐近时间跨度最优性保证。此外，当引入规则分布的障碍物时，我们的方法性能不会下降。这些方法可推广至支持$100\%$的智能体密度。无论维度与密度如何，我们高质量的方法得益于两个关键构建模块的独特层次化集成。在高层级，我们应用了带标签的网格重排算法（RTA），该算法能够通过行/列混洗在网格上执行高效重构。在低层级，我们设计了新颖方法以高效模拟RTA返回的行/列混洗操作。我们基于RTA算法的实现方案在广泛的数值评估中表现出高效性，相较于其他最先进方法展现出优异的可扩展性。例如，在三维场景中，基于\rta的算法可轻松扩展至超过$370,000$个顶点和超过$120,000$个智能体的网格，并持续实现接近$1.5$的保守时间跨度最优性，这与我们的理论分析预测一致。