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 A[i, 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 | beer | 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 a 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?
Show that Diverse Subset is NP-complete.
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