0/1 knapsack problem: Greedy Algorithm Counterexample

While reading about 0/1 knapsack problem on the Internet, many tutorials considered value/weight ratio to solve the problem and I was wondering will it always contain the element with the greatest value/weight ratio to find an optimal solution to a 0/1 knapsack problem or is there any example where we find the optimal solution without the greatest ratio?

Consider this counterexample. Suppose the knapsack has a capacity 4. And suppose there are three items:

• Item A with weight 3 and value 5

• Item B with weight 2 and value 3

• Item C with weight 2 and value 3

The optimal solution contains items: B and C, in the knapsack with a total value = 6. However, item A has the highest value/weight ratio: 5/3, which is greater than 3/2.

StackExchange Computer Science Q#141286, answer score: 4