Maximize sum of table where each number must come from unique row and column

Question

Suppose we have a table of numbers like this (we can assume it is a square table):

20  2   1   3   4
5   1   14  8   9
15  12  17  17  11
16  1   1   15  18
         


        

Answer 1

This is the maximum cost bipartite matching problem. The classical way to solve it is by using the Hungarian algorithm.

Basically you have a bipartite graph: the left set is the rows and the right set is the columns. Each edge from row i to column j has cost matrix[i, j]. Find the matching that maximizes the costs.