Find the minimum amount of swaps to sort array

Your question appears to be about sorting networks. Sorting an array in the comparison model requires $Omega(nlog n)$ comparisons, and so $Omega(nlog n)$ of your swaps. Ajtai, Komlós and Szemerédi were the first to come up with a matching $O(nlog n)$ sorting network (the AKS sorting network), and their construction was simplified by Patterson. These networks also have the advantage that they can be divided into $O(log n)$ layers of disjoint swaps. Very recently, Goodrich came up with Zigzag sort, another $O(nlog n)$ sorting network. Since we know that there exist $O(nlog n)$ sorting networks, we can find an optimal sorting network in time $binom{n}{2}^{O(nlog n)} = 2^{O(nlog^2 n)}$ (verifying that a network works takes time roughly $2^n$ using the zero-one principle). There is no reason to expect any subexponential algorithm. You might be interested in Ian Parberry’s page on sorting. This part answers the following question: What is the maximal number of swaps needed to order an array of length $n$? Suppose that your array contained numbers from $1$ to $n$. Then you can think of it as a permutation $pi in S_n$. Swapping two elements in the same as multiplying by a transposition, so the question is how many transpositions we need to multiply to get $pi$. If the cycle structure of $pi$ is $a_1,ldots,a_k$ then this number is $(a_1-1) + cdots + (a_k-1) = n – k$. Therefore $n-1$ is the most that is needed. An example of a permutation needing $n-1$ swaps is $(234cdots n1)$, which corresponds to the array $2,3,4,ldots,n,1$.