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
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This note was uploaded on 12/13/2010 for the course CSCI 570 at USC.