What is the fastest way to select $n$ values from an $n$ by $n$ matrix such that each value comes from a different row and column and the sum of those n values is minimized?
For example, given the matrix:
$$A=\begin{bmatrix}0 & 1 &5 \\1 & 4 & 3\\2 & 7 & 0\end{bmatrix}$$
The solution is $\{A_{1,2}, A_{2,1}, A_{3,3}\}$, taking 1 from the first row, 1 from the second row, and 0 from the third row.
In the brute-force case, there are $n!$ possible solutions that have to be analyzed, so the ovreall running time is $O(n!)$, although are there any faster algorithms to solve this?
