Sort array with minimum swaps
Let $A$ be the array; by hypothesis there exists a unique permutation $\pi : A \rightarrow A$ such that $\pi(A)$ is sorted. The minimum number of swaps required to sort $A$ is the minimum $h$ such that there exists a decomposition of $\pi$ in $h$ transpositions. 

Certainly $\pi$ can be broken down in at most $n-k$ transpositions where $k$ is the number of cycles in the unique cycle decomposition of $\pi$: it is sufficient to observe that a cycle $(c_1, c_2, \dots, c_l)$ can be rewritten as $(c_1, c_l)(c_1, c_{l-1})\dots(c_1, c_2)$.

It can also be shown that $n-k$ is the minimum, the proof isn't straightforward but it's a known result that you can easily find in literature should you be interested.

As for your multi-dimensional case, the same idea should work, but you should be more specific about what you mean for a multi-dimensional array to be sorted.