Anyone knows an algorithm that finds minimum distance for all permutations?
Surprisingly, if the distance is $L_p$, or the $p$-norm induced metric, then you simply sort $A$ and $B$ respectively, and then compute their distance, and that distance will be guaranteed minimal over all permutations. That is time complexity of $O(n\log n)$ and I believe it is the most efficient one.

The key proof is done in https://math.stackexchange.com/questions/1984686/is-x-1-y-2px-2-y-1p-ge-x-1-y-1px-2-y-2p-for-any-0-le-x-1-le.

Sometimes, before setting out to find an "efficient" algorithm, it might be a good practice to first think of proving the "greedy" algorithm is not right; because if the greedy indeed works, all efforts to search for the "efficient" algorithm will be in vain.