We investigate the multi-agent pathfinding (MAPF) problem with n agents on graphs with n vertices: Each agent has a unique start and goal vertex, with the objective of moving all agents in parallel movements to their goal s.t. each vertex and each edge may only be used by one agent at a time. We give a combinatorial classification of all graphs where this problem is solvable in general, including cases where the solvability depends on the initial agent placement. Furthermore, we present an algorithm solving the MAPF problem in our setting, requiring O(n2) rounds, or O(n3) moves of individual agents. Complementing these results, we show that there are graphs where Ω(n2) rounds and Ω(n3) moves are required for any algorithm.

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