Question

In this assignment, you will prove that the Zero Sum problem is NP-Complete. The Zero Sum problem accepts an array of n integers and returns whether it is possible to negate some of those integers to make the array sum equal 0. For example, the array data = [1, 3, 5, 7, 9, 11] has a zero sum if you negate the 7 and 11, while the array data = [1, 2, 5, 9, 13, 22] cannot be reduced to a sum of zero.

1. Prove that Zero Sum is NP.

2. The Subset Sum problem is NPC. This takes in an array of n integers and a target t to determine if the array has a subset that sums to t. To prove that Zero Sum is NPC, would you reduce Zero Sum to Subset or Subset Sum to Zero Sum?

3. Give pseudocode for a reduction in the appropriate direction. NOTE: This answer is based on the CORRECT answer for #2. Subset Sum has two parameters (an array and a target value) while Zero Sum only has one (an array). If you are reducing ZS to SS, you will need to come up with a target value and possibly modify the array. If you are reducing SS to ZS, you will need to come up with an array that incorporates the target value.

4. Show that your reduction is polynomial time

Answer 1

1. To show that Zero Sum is NP we need to show that we can verify a certificate of Zero Sum in polynomial time.
This can be done easily, as we can take the indices of the elements of the array which need to be negated as our certificate. Now, to verify if the certificate is correct, we just need to find the sum of the elements of the array after negating the elements in the corresponding indices of the array and return True if the sum is 0 or False otherwise.

2. We need to reduce Subset Sum to Zero Sum in polynomial time to prove that Zero Sum is NP complete.

3. Let S be an array and t be a target value. We need to find a reduction f such that f(S) will be in Zero Sum if and only if <S,t> is in Subset Sum. In other words, we find an array negating sum of whose elements and taking the sum of all the elements will give 0 if and only if there is a subset of S whose sum of elements is t.

We can form the power set of the given array S, and append the element -t to each of the elements in the power set. We now need to find if any of the elements of the power set belongs to Zero Sum.

4. Though the number of elements in the power set of S is , the power set can be formed in polynomial time. So, the reduction is polynomial.