Give a family of set cover problems where the set to be covered has n elements, the minimum set cover is size k=3, and the greedy algorithm returns a cover of size (logn). That is, you should give a description of a set cover problem that works for a set of values of n that grows to infinity you might begin, for example, by saying, Consider the set X={1,2,...,2^b} for any b 10, and consider subsets of X of the form..., and finish by saying We have shown that for the example above, the set cover returned by the greedy algorithms of size b = (logn). (Your actual wording may differ substantially) Explain briefly how to generalize your construction for other (constant) values of k. (You need not give a complete proof of your generalization, but explain the types of changes needed from the case of k=3.)

Time complexity: it cannot be exponential, but there are no strict expectations of time complexity. It just cannot be terribly inefficient.

Edit: No set information is explicitly given. We can determine our own set constraints, for example a subset that consists of all prime numbers, a subset where each element is divisible by 10, and a subset of all even numbers.