Counterexample - The Classic

Here is the classic counterexample: Items: (weight= P0, value= P1), (weight= P2, value= P3), (weight= P4, value= P5). Capacity = $50$. Ratios: $6$, $5$, $4$. Greedy takes item $1$ (ratio P11), then item $2$ (ratio P13). Total weight: $30$, total value: $160$. Item $3$ does not fit (need P17 more, only P18 left).

But items $2$ + $3$: weight = $50$, value = $220$. Much better! Greedy got $160$, optimal is $220$. That's why $0$/$1$ knapsack needs DP. This example shows greedy's local choice blocks a better global solution.