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

For starters, you can use dynamic programming.

In your straightforward approach, you are doing exactly the same computation many, many times.

For example, at some point you answer the question: "For the last three columns with rows 1 and 2 already taken, how do I maximize the sum?" You compute the answer to this question twice, once when you pick row 1 from column 1 and row 2 from column 2, and once when you pick them vice-versa.

So don't do that. Cache the answer -- and also cache all similar answers to all similar questions -- and re-use them.

I do not have time right now to analyze the running time of this approach. I think it is O(2^n) or thereabouts. More later maybe...

Answer 2

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.