The computational complexity of multi-agent pathfinding on directed graphs

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.