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HW4 - Homework#4 Due Date Wednesday October 27th start of...

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Homework #4 Due Date: Wednesday, October 27th, start of class This is a shorter homework which you are not expected to seriously work on until after the midterm exam. However, it is recommended you attempt problem #1, which is of approximate difficulty to what you will encounter on the midterm. 1. In the Half-Cover problem, we are given m sets S 1 , S 2 , ..., S m , each of which contains a subset of the integers 1 , 2 , ..., n . Our goal is to determine whether there exists a collection of k sets whose union has size at least n 2 . (a) Suppose we prove that Half-Cover is NP-complete, and that we find an O ( n 4 ) algorithm for Half-Cover. Does this imply that there is a polynomial algorithm for 3-SAT? Does this imply that there is an O ( n 4 ) 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 graph G = ( V, E ), an integer k , and a target T . We want to divide the nodes into k sets such that any pair of nodes in the same set have a path of length T to each other. Prove that this problem is NP-complete.
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