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6. The Course Offering Problem is: Given an integer K and a set of n courses and a set of m students and, for each student si

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Answer #1

Answer to (a):

First to prove the problem is in NP, we need to find a efficient cirtifier for the problem. So there exist a cirtifier such that for any yes instance, the cirtifier will accept that cirtificate. Similiarly the cirtifier will not have any cirtificate for any no instance of the problem. The running time of the cirtificate must be in polynomial time.

Now, given a cirtificate as the set of courses offered to the students, the cirtifier can give a cirtificate depending on the instance. Given a collection of set of courses S, a cirtifier can check if it contains atmost k elements and union of all elements forms the universal set U.

Answer to (b):

The Set Covernf problem states: Given a universal set U and a family S of subsets of U, a cover is a subfamily C\subseteqS of sets whose union is U. The input is a pair (U,S) and an integer k, the question is whether there is a set covering of size k or less.

It is exactly like the Course offering problem. U represents the set of all classes available. The input is instance of courses offered to the students and all the courses. k is a integer such that it is possible to offer k courses.

So the Set Covering problem can be reduced to Course Offering Problem in polynomial time.

So, Course offering problem is also NP Complete.

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