For practice with LinearProgram, look at LPExamples.
1. Give an approximation algorithm for the following problem, a generalization of set cover.
The input is a set of elements E = {e1, e2,…, en}, a set of sets {S1, S2, …, Sm}, and an integer budget B.
Given a collection of the sets, define the coverage of the collection to be the minimum, over all elements e, of the number of sets in the collection containing e.
A solution is a collection of B sets that maximizes the coverage.
Describe a randomized approximation algorithm that, with probability at least 1/2, finds a solution with coverage at least OPT - sqrt(2 OPT log (2n)), where OPT is the maximum coverage of any collection of B of the sets.