Traditionally the hungarian algorithm is used to solve the assigment problem. Later on, more efficient or simpler algorithms were deviced such as Mack's algorithm. This is a swift implementation of this algorithm.
More details about this algorithm can be found in this pdf.
This is an example how to program works. Suppose the following 12x12 cost matrix is given:
(in red are the obvious minimums per row)
Then the algorithm will output all intermediate results like this:
And eventually will output the following optimal assigment:
To see the colors in the console window of Xcode as shown above you'll need to install a special font called ColoredConsole-Bold.ttf. You can download this font and find instruction on how to install it in Xcode on this github repo.