In this paper, we study the structure and computational complexity of optimal multi-robot path planning problems on graphs. Our results encompass three formulations of the discrete multi-robot path planning problem, including a variant that allows synchronous rotations of robots along fully occupied, disjoint cycles on the graph. Allowing rotation of robots provides a more natural model for multi-robot path planning because robots can communicate. Our optimality objectives are to minimize the total arrival time, the makespan (last arrival time), and the total distance. On the structure side, we show that, in general, these objectives demonstrate a pairwise Pareto optimal structure and cannot be simultaneously optimized. On the computational complexity side, we extend previous work and show that, regardless of the underlying multi-robot path planning problem, these objectives are all intractable to compute. In particular, our NP-hardness proof for the time optimal versions, based on a minimal and direct reduction from the 3-satisfiability problem, shows that these problems remain NP-hard even when there are only two groups of robots (i.e. robots within each group are interchangeable).