Integer Knapsack Problem - No duplicates Allowed
You are right about "independence" between subproblems. Therefore, you use the new subproblems (wiki) by introducing another variable $i$, meaning using the items up to $i$:


  Define $V[i,s]$ to be the maximum value ($V$) that can be attained with size less than or equal to $s$ using items up to $i$ (the first $i$ items).