4. [30 points] Collaborative Problem: We start by defining the Independent Set problem. Given a graph G = (V,E), we say a set of nodes S-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 indepen- dent set that is as large as possible. Stated as a decision problem, IS answers the question, does there exist a set S EG such that |S| 2 k? Then set k as large as possible. For this problem, you may take as given that IS is NP-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 colums (or fields) correspond to products the store sells. The entry Ai,) specifies the quantity of product j that has been purchased by customer i. For example, Table 1 shows one such table Table 1: Customer Tracking Table CustomerDetergent Beer Diapers Cat Litter Raj Alanis Chelsea 0 0 0 0 0 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 customres in S have ever bought the same product (i.e., for each product, at most 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 n array A as de- fined above and a number k