Max Value to True/False

In classic knapsack, $dp[w]$ stores the maximum value achievable with capacity $w$. But some problems don't ask for maximum value. They ask: "Can you make exactly this sum?" For these problems, $dp[s]$ becomes a boolean: true if sum $s$ is achievable, false otherwise. The formula changes too.

Instead of $dp[w] = \max(dp[w], dp[w-wt] + val)$, you get $dp[s] = dp[s] \text{ OR } dp[s - num]$. 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.