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CSCI 256
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Only turn in problems from the second section.
Please do problem 36.2-8 on page 929 in CLR.
Please do problem 36.3-1 on page 938 in CLR.
Please do problem 36.3-7 on page 939 in CLR.
Consider the following algorithm to determine whether a graph has a clique of size k. First generate all subsets of the vertices containing exactly k vertices. There are "n choose k" or O(nk) subsets of size k altogether. Then, we check whether any of the subgraphs induced by these subsets is complete. Why is this not a polynomial algorithm for the clique problem, which implies that P = NP?
Show by reduction from CLIQUE that the INDEPENDENT SET problem is NP-complete. (That is, you may assume that CLIQUE is NP-complete in your proof.)
INDEPENDENT SET: An independent set in a graph G = (V,E) is a set of vertices, no two of which are connectied. The problem is to determine, given G and an integer k, whether G contains an independent set with >= k vertices.
Please do problem 36.2-5 in CLR. Hint: How can you get the needed certificate?
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