Given $n$ objects $o_i$ with weight $w_i$ and value $v_i$, and a goal $g$ is there a bag that has weight less than $B$ but value at least $g$. The solution to this is a subset of objects $S \subset \{1, 2, \ldots, n\}$ such that:

• $\sum_{i \in S} w_i \leq B$, and
• $\sum_{i \in S} v_i \geq g$.

If there is such a bag what is it otherwise say there is no such solution.