Maximise sum of “non-overlapping” numbers from matrix

Question

Just looking for a bit of direction, I realise that the example given is possible to solve using brute force iteration, but I am looking for a more elegant (ie. mathematical

Answer 1

As Matthias indicated you should use backtracking.

1. Find a reasonable solution. Select max values from each row and see if they are non-overlapping. If not, then perturb part of the solution so that the result becomes non-overlapping.
2. Define fitness of a partial solution. Let us say you are picking up value for each row iteratively and you have already picked values from first k rows. The fitness of this solution equals sum of the already picked values + max values from remaining rows and unselected columns
3. Now recursively start searching for solution. Select the values from first row, calculate their fitness and insert them into a priority queue. Remove all the solutions whose fitness is lower than the current optimal solution (initialized in step 1). Pick the solution at the head of the queue, calculate the next level of solutions and insert them back to the priority queue. Once you have selected values from all columns and rows, calculate the sum, and if it is higher than current optimal, replace it.