A store trying to analyze the behavior of its customers will often maintain a two-dimensional array A, where the rows correspond to its customers and the columns correspond to the products it sells. The entry Ali, j] specifies the quantity of product j that has been purchased by customer i. Here's a tiny example of such an array A liquid detergent bee diapers cat litter
Raj 0 6 0 3
Alanis 2 3 0 0
Chelsea 0 0 0 7
One thing that a store might want to do with this data is the following. Let us say that subset S of the customers is diverse if no two of the of the customers 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 as follows:
Given an m × n array A as defined above, and a number k < m, is there a subset of at least k of customers that is diverse? 1. Show that Diverse Subset is in NP.
2. Show that Diverse Subset is NP-complete