What is the difference between 0/1 knapsack problem and fractional knapsack problem?

The 0/1 knapsack problem and the fractional knapsack problem are two variations of the classic knapsack problem in algorithm design. The main difference between them lies in the constraints and the nature of the items that can be included in the knapsack.

1. 0/1 Knapsack Problem:
In the 0/1 knapsack problem, each item is either included entirely in the knapsack or not included at all. This means that the items are indivisible, and you cannot take a fraction of an item. The objective is to maximize the total value of the items in the knapsack while ensuring that the total weight does not exceed the knapsack's capacity.

The 0/1 knapsack problem is known for its combinatorial nature, as it requires making binary decisions for each item. This problem is typically solved using dynamic programming techniques, such as the 0/1 knapsack algorithm, which builds a table to store the optimal solutions for subproblems.

2. Fractional Knapsack Problem:
In the fractional knapsack problem, items can be included in fractions or portions. This means that you can take a fraction of an item, allowing for more flexibility in selecting items. The objective is still to maximize the total value of the items in the knapsack, but now the total weight can exceed the knapsack's capacity.

The fractional knapsack problem is often solved using a greedy algorithm approach. The algorithm sorts the items based on their value-to-weight ratio and then selects items in a greedy manner, starting with the highest ratio. This approach ensures that the most valuable items are chosen first until the knapsack is full.

In summary, the main difference between the 0/1 knapsack problem and the fractional knapsack problem is the divisibility of items. The 0/1 knapsack problem only allows for binary decisions, while the fractional knapsack problem allows for fractions or portions of items to be included. This difference in divisibility affects the algorithms used to solve each problem, with dynamic programming commonly used for the 0/1 knapsack problem and a greedy algorithm for the fractional knapsack problem.