What is the Subset Sum problem and how can it be solved using Dynamic Programming?

The Subset Sum problem is a computational problem that involves finding a subset of a given set of integers whose sum equals a given target value. The problem can be stated as follows: given a set of positive integers and a target sum, determine whether there is a subset of the given set whose sum is equal to the target sum.

Dynamic Programming can be used to solve the Subset Sum problem efficiently. The problem can be broken down into smaller subproblems, where we consider subsets of the given set and their corresponding sums. By solving these subproblems and storing their results in a table, we can build up to the solution of the original problem.

The dynamic programming approach involves creating a 2D table, where the rows represent the elements of the given set and the columns represent the possible target sums. The table is initialized with False values. We then iterate through each element of the set and each possible target sum, filling in the table based on the following rules:

1. If the target sum is 0, then the answer is True, as an empty subset can be formed with a sum of 0.
2. If the current element is greater than the target sum, then we can't include it in the subset, so the value in the table remains the same as the previous row.
3. If the current element is less than or equal to the target sum, we have two options:
a. Exclude the current element and check if the subset sum can be achieved without it, i.e., check the value in the previous row for the same target sum.
b. Include the current element and check if the subset sum can be achieved by subtracting the current element from the target sum, i.e., check the value in the previous row for the target sum minus the current element.
If either of these options is True, then the value in the table for the current element and target sum is set to True.

After filling in the entire table, the value in the bottom-right cell represents whether a subset with the target sum exists in the given set. If it is True, then a subset with the target sum can be formed; otherwise, it is not possible.

Overall, the dynamic programming approach for the Subset Sum problem has a time complexity of O(n*sum), where n is the number of elements in the set and sum is the target sum.