  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.
