For instance, is it legal to try to do a tour of the matrix, to pick up as many high value cells as possible? If the priority is "max sum", followed by "min terms", then a "rook's tour" of the positive valued cells is in order.
Another idea, can you revisit a cell? What's to keep you from circling through 4 positive value cells forever?
Can you go "offline", by taking a longer tour than necessary?
Quantum Mechanics: The dreams stuff is made of