While the non-optimizing variant of multi-agent pathfinding on undirected graphs is known to be a polynomial-time problem since almost forty years, a similar result has not been established for directed graphs. In this paper, it will be shown that this problem is NP-complete. For strongly connected directed graphs, however, the problem is polynomial. And both of these results hold even if one allows for synchronous rotations on fully occupied cycles. Interestingly, the results apply also to the so-called graph motion planning feasibility problem on directed graphs.

中文翻译：

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有向图上多智能体寻路的计算复杂度

虽然近四十年来无向图上多智能体寻路的非优化变体被认为是多项式时间问题，但对于有向图尚未建立类似的结果。本文将证明该问题是NP完全问题。然而，对于强连接的有向图，问题是多项式的。即使允许在完全占用的周期上进行同步旋转，这两个结果仍然成立。有趣的是，结果也适用于有向图上所谓的图运动规划可行性问题。