Multi-Agent Path Finding (MAPF) is a fundamental motion coordination problem arising in multi-agent systems with a wide range of applications. The problem's intractability has led to extensive research on improving the scalability of solvers for it. Since optimal solvers can struggle to scale, a major challenge that arises is understanding what makes MAPF hard. We tackle this challenge through a fine-grained complexity analysis of time-optimal MAPF on 2D grids, thereby closing two gaps and identifying a new tractability frontier. First, we show that 2-colored MAPF, i.e., where the agents are divided into two teams, each with its own set of targets, remains NP-hard. Second, for the flowtime objective (also called sum-of-costs), we show that it remains NP-hard to find a solution in which agents have an individually optimal cost, which we call an individually optimal solution. The previously tightest results for these MAPF variants are for (non-grid) planar graphs. We use a single hardness construction that replaces, strengthens, and unifies previous proofs. We believe that it is also simpler than previous proofs for the planar case as it employs minimal gadgets that enable its full visualization in one figure. Finally, for the flowtime objective, we establish a tractability frontier based on the number of directions agents can move in. Namely, we complement our hardness result, which holds for three directions, with an efficient algorithm for finding an individually optimal solution if only two directions are allowed. This result sheds new light on the structure of optimal solutions, which may help guide algorithm design for the general problem.

翻译：多智能体路径规划（Multi-Agent Path Finding, MAPF）是一个基础的运动协调问题，源于多智能体系统，具有广泛的应用。该问题的难解性促使研究者广泛探索如何提升求解器的可扩展性。由于最优求解器可能难以扩展，关键挑战在于理解MAPF的困难根源。我们通过细粒度复杂度分析，研究二维网格上的时间最优MAPF问题，从而填补两项空白，并识别出新的可处理性前沿。首先，我们证明双色MAPF问题（即智能体分为两队，每队有各自的目标集）仍然是NP难的。其次，针对流程时间目标（也称为总成本最小化），我们证明寻找所有智能体都达到个体最优成本的解（称为个体最优解）仍然是NP难的。此前针对这些MAPF变体的最紧密结果仅适用于（非网格）平面图。我们采用单一难度构造，替代、强化并统一了之前的证明。我们认为该构造比平面情况下的先前证明更简洁，因为它使用最小化组件，能在单张图中完全可视化。最后，针对流程时间目标，我们基于智能体的可移动方向数量建立了可处理性前沿。具体而言，我们补充了适用于三个方向的难度结果，并提出了一个高效算法，在仅允许两个方向时寻找个体最优解。这一结果揭示了最优解结构的新见解，可能有助于指导一般问题的算法设计。