To prove that the set partition problem is NP complete.

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Chapter 34.5, Problem 5E

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To prove that the set partition problem is NP complete.

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A graduate student is working on a problem X. After working on it for several days she is unable to find a polynomial-time solution to the problem. Therefore, she attempts to prove that he problem is NP-complete. To prove that X is NP-complete she first designs a decision version of the problem. She then proves that the decision version is in NP. Next, she chooses SUBSET-SUM, a well-known NP-complete problem and reduces her problem to SUBSET-SUM (i.e., she proves X £p SUBSET-SUM). Is her approach correct? Explain your answer.

The subset sum problem is defined as follows: SUBSETSUM = {(a1, a2,...,am; b) : m,a1, a2, ...,am; b are integers and 3IC {1,2, ..., m} such that Eier a; =b}. Assume you have a polynomial-time algorithm A that decides, for any input sequence (a1, a2, ...,am, b), whether or not (a1, a2, ...,am, b) E SUBSETSUM. Note that this algorithm only returns YES or NO; it does not return anything else. Design a polynomial-time algorithm B that takes an arbitrary sequence (a1, az, ..., am, b) as input. • If (a1, a2, ...,am; b) E SUBSETSUM, then B returns a subset I of {1,2, ..., m} such that Eier a; = b. If (a1, a2, ..., am, b) & SUBSETSUM, then B returns NO. Your algorithm B may use algorithm A as a black box. As always, justify your answer.

et X = {1, 2, 3, 4, 5} and Y = {7, 11, 13} are two sets. find R = {(x, y): x ∈ X and y ∈ Y and (y – x) is divisible by 6}