In classic knapsack, stores the maximum value achievable with capacity . But some problems don't ask for maximum value. They ask: "Can you make exactly this sum?" For these problems, becomes a boolean: true if sum is achievable, false otherwise. The formula changes too.
Instead of , you get . This boolean version is called subset sum DP. It's simpler than value-based knapsack but appears everywhere: partition problems, target sum, even some game theory problems.