We study the problem of optimal multi-robot path planning on graphs (MPP) over four distinct minimization objectives: the total arrival time, the makespan (last arrival time), the total distance, and the maximum (single-robot traveled) distance. On the structure side, we show that each pair of these four objectives induces a Pareto front and cannot always be optimized simultaneously. Then, through reductions from 3-SAT, we further establish that computation over each objective is an NP-hard task, providing evidence that solving MPP optimally is generally intractable. Nevertheless, in a related paper, we design complete algorithms and efficient heuristics for optimizing all four objectives, capable of solving MPP optimally or near-optimally for hundreds of robots in challenging setups.
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