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

Your problem is almost identical to the Assignment problem, which can e.g. be solved by the Hungarian algorithm in polynomial time.

Note that the assignment problem is usually a minimization problem, but multiplying your matrix with -1 and adding some constant should make the method applicable. Further, there is no formal tie-braking condition, for case of multiple optimal solutions. However, the method yields you a solution having the optimal sum. Let m be the minimum summand. Modify your matrix by setting all entries less or equal to m to zero and solve again. Either you get a solution with the same sum that is better than the last one. If not, the previous solution was already optimal.