Lecture 9

Learning objectives

After this class, you should be able to:

1. Given an instance of the set cover problem, show the steps involved in obtaining an approximate solution using the primal-dual algorithm given in class.
2. Prove that the above algorithm yields a feasible solution, and determine its approximation factor.
3. Show how we can express the above algorithm as a purely combinatorial algorithm, without referring to linear programming.

Reading assignment

1. AA: Chapter 15.
2. AA: Chapter 16, page 130.

Exercises and review questions

• Questions on current lecture's material
  1. Solve the following set cover problem using the primal-dual algorithm discussed in class. Show the steps. Also, give the value of f for this problem. Determine the optimal solution too, and show that the ratio of approximate solution to the optimal solution is consistent with theorem 15.3. Instance: sets {a, b, c}, {a, c}, {a, b, e}, and {b, d} with weights 3, 2, 3, and 2 respectively.
  2. Explain AA: Example 15.4.
  3. AA: Exercise 15.2: Remove the scaffolding of linear programming from algorithm 15.2, to obtain a purely combinatorial factor f algorithm for set cover.

• Questions on next lecture's material
  1. (Post your answer on the discussion board) Give an instance of the MAX-SAT problem, and a feasible solution for that instance.