Solution

The correct answer is 3)(A), (C), (D), (E), (B).

Key Points

• Define the base case when the capacity is zero (0) or no items are left to consider: This step ensures that we have a stopping condition for our recursive algorithm. If the knapsack's capacity is zero or there are no items left to consider, the maximum value is zero.
• Identify subproblems and their dependencies based on items weights and values: This involves breaking down the main problem into smaller subproblems that can be solved independently. The dependencies will be based on the weights and values of the items.
• Initialize a table to store results of subproblems: A table (usually a 2D array) is initialized to store the results of these subproblems. This helps in avoiding the recomputation of the same subproblems and thus improves efficiency.
• Iterate through each item and each possible Capacity to fill the table: This step involves populating the table based on the subproblems identified. For each item and each possible capacity, we decide whether to include the item in the knapsack or not.
• Compute the maximum value that can be obtained using items up to the i-th item and a knapsack capacity of 0: Finally, compute the maximum value from the filled table. This gives the solution to the original problem.

Thus the correct answer is ​(A), (C), (D), (E), (B).

Additional Information

• The Knapsack problem is a classic example of a combinatorial optimization problem, and dynamic programming provides an efficient way to solve it.
• Understanding the structure of the problem and the dependencies between subproblems is crucial for correctly implementing the dynamic programming solution.
• The complexity of the dynamic programming approach is generally O(nW), where n is the number of items and W is the capacity of the knapsack. This is much more efficient than the exponential time complexity of the naive recursive solution.