Consider an optimization version of the Hitting Set Problem defined as follows. We are giv...

Consider an optimization version of the Hitting Set Problem defined as follows. We are given a set A = {a1, an} and a collection B1, B2,..., Bm of subsets of A. Also, each element ai e A has a weight wt > 0. The problem is to find a hitting set H c A such that the total weight of the elements in H, that is, is as small as possible. (As in in Chapter 8, we say that H is a hitting set if H ∩ Bt is not empty for each i.) Let b = maxi |Bi| denote the maximum size of any of the sets B1, B2,... ,Bm. Give a polynomial-time approximation algorithm for this problem that finds a hitting set whose total weight is at most b times the minimum possible.