Knapsack algorithm with an additional property

Question

When there\'s 1 property, I do understand what\'s going on in there. I\'m having a problem with understanding knapsack problem when there\'s more than 1 property.

Answer 1

Say you are using a three dimension table: A[x][y][z]=k, x: sum 1st property; y: sum 2nd property; z: sum 3rd property; k: minimal cost (maximal reward, which I prefer using reward)

So you iterate over items. Say current item is (p1, p2, p3, reward) (reward = - cost). for each (x,y,z,k), your update formula:

A[x+p1][y+p2][z+p3] = max(A[x+p1][y+p2][z+p3], A[x+p1][y+p2][z+p3] + reward)

If the 1st term on RHS is greater, on slot A[x+p1][y+p2][z+p3], the configuration of knapsack is remain still; otherwise, you update the knapsack by that of A[x+p1][y+p2][z+p3] plus the current item.

Hope this cut things clear.