[30 points] Collaborative Problem: We start by defining the Independent Set Problem (IS). Given a graplh G(V, E), we say a set of nodes S C V is independent if no two nodes in S are joined by an edge. The Independent Set Problem, which we denote IS, is the following. Given G, find an independent set that is as large as possible. Stated as a decision problem, IS answers the question: "Does there exist a set s s V such that isl 2 ?Then set k as large as possible. For this problem, you may take as given that IS is N P-complete. A store trying to analyze the behavior of its customers will often maintain a table A where the rows of the table correspond to the customers and the columns (or fields) correspond to products the store sells. The entry Ali, jl specifies the quantity of product j that has been purchased by customer i. For example, Table ?? shows one such table. One thing that a store might want to do with this data is the following. Let's say that a subset S of the customers is diverse if no two of the customers in S have ever bought the same product (i.e., for each product, at most Table 1: Customer Tracking Table Customer Detergent Beer Diapers Cat Ltter Rai anis Chelsea one of the customers in S has ever bought it). A diverse set of customers can be useful, for example, as a target pool for market research. We can now define the Diverse Subset Problem (DS) as follows: Given an m x n array A as defined above and a number k s m, is there a subset of at least k customers that is diverse? Prove that DS is NP-complete