In Exercise 1 of we claimed that the Hitting Set Problem was NP-complete. To recap the def...

In Exercise 1 of we claimed that the Hitting Set Problem was NP-complete. To recap the definitions, consider a set A= {a1, …, an} and a collection B1, B2, … , Bm of subsets of A. We say that a set H ⊆ A is a hitting set for the collection B1, B2, . . . , Bm if H contains at least one element from each Bi–that is, if H ∩ Bi is not empty for each i. (So H "hits" all the sets Bi.)

Now suppose we are given an instance of this problem, and we'd like to determine whether there is a hitting set for the collection of size at most k. Furthermore suppose that each set Bt has at most c elements, for a constant c. Give an algorithm that solves this problem with a running time of the form O (f  (c, k) • p (n, m)), where p (•) is a polynomial function, and f  (•) is an arbitrary function that depends only on c and k, not on n or m.

Exercise 1

For each of the two questions below, decide whether the answer is (i) "Yes," (ii) "No," or (iii) "Unknown, because it would resolve the question of whether ." Give a brief explanation of your answer.

(a) Let's define the decision version of the Interval Scheduling Prob­lem from Chapter as follows: Given a collection of intervals on a time-line, and a bound k, does the collection contain a subset of nonoverlapping intervals of size at least k?

Question: Is it the case that Interval Scheduling ≤P Vertex Cover?

(b) Question: Is it the case that Independent Set ≤P Interval Scheduling?