Multi agent path finding: NP-hard or not?

Question

The question is as such: A 2d grid with obstacles has been given with 'n' robots. The robots can move in the adjacent cell within one time-tick. There are multiple constraints on robots like they cannot be in the same cell at the same time. However, they can move simultaneously. An action plan needs to be found out for each robot that moves the robot from a starting point to destination such that constraints are satisfied and the overall total time taken by the robots is the least.

I cannot seem to grasp whether this problem is an NP-hard problem or not

Answer 1

A quick search turns up a recent paper on this topic:

On the Computational Complexity ofMulti-Agent Pathfinding on Directed Graphs

The problem is NP hard in this special case of directed graphs. There are polynomial time algorithms for finding solutions in many cases, but finding optimal solutions is NP hard even on undirected graphs. (The Sliding Tile Puzzle is NP-Hard, and multi-agent pathfinding is a special case.)