Homework #4Due Date:Wednesday, October 27th, start of classThis is a shorter homework which you are not expected to seriously work on until after themidterm exam. However, it is recommended you attempt problem #1, which is of approximatedifficulty to what you will encounter on the midterm.1. In the Half-Cover problem, we are givenmsetsS1, S2, ..., Sm, each of which contains a subsetof the integers 1,2, ..., n. Our goal is to determine whether there exists a collection ofksetswhose union has size at leastn2.(a) Suppose we prove that Half-Cover is NP-complete, and that we find anO(n4) algorithmfor Half-Cover. Does this imply that there is a polynomial algorithm for 3-SAT? Doesthis imply that there is anO(n4) algorithm for 3-SAT? Explain your reasoning.(b) Prove that Half-Cover is NP-complete.2. In the Clustering problem, we are given a weighted graphG= (V, E), an integerk, and atargetT. We want to divide the nodes intoksets such that any pair of nodes in the sameset have a path of length≤Tto each other. Prove that this problem is NP-complete.
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Natural number, Prime number, Universal quantification