OK. But something does not add up: You can slide over negative values, essentially ignoring them. They have no effect whatsoever! right? Oh except when they are on a turning.

Here are some more questions to rubber-duck it

Do you know your matrix fully in advance? Or do you have to query it as you move along? Because it may be too expensive to evaluate or store the full matrix, for example evaluating a function in 10-decimal accuracy.

Do you have to find the best solution or can you use a heuristic like genetic algorithms which may find a good solution in short time by skipping some (possibly better) solutions? If so, one simple heuristic is to start first with those squares with large points and slide through them. Or plan a path which includes most valueable squares and no negatives. Completely ignoring the small ones and hoping for the best! More concretely, here is a top-down rather than bottom-up approach: first sort your squares wrt value and this will determine your visitation priority. Then construct a path-builder to suggest paths given these top-priority, must-visit squares.

Btw, it reminds me of Gradient Descent (or Hill Climbing). It still suffers from local minima but there are some better alternatives out there fo exploring a neighbourhood and avoiding local minima ...

In any case I thought perhaps you can cache path segments you visit so that you do not have to re-sum.

bw, bliako

1) exactly ! I only cannot skip negative's or 0's if i am on corners

2) matrix is static fully known in advance (real matrix is 50000*50000 32bit int) so any preprocessing is more than welcomed
problem is that every time i get two different random values for which i need to compute the best solution. Optimal solution would be ok if it is within some proven cutoff (bound) from the best one (I would need to provide some guarantees if deciding to go for a suboptimal solution)

I'll check into the stated potions and repost my solution so maybe someone can do something better

Thank you !!!

This looks a lot like a graph theoretical problem with the incidence matrix representing the weights of a directed graph.

HTH!

Cheers Rolf
_{(addicted to the Perl Programming Language :)
Wikisyntax for the Monastery}