Lecture 7

Learning objectives

After this class, you should be able to:

1. Define the terms: (i) primal program, (ii) dual program, and (iii) integrality gap.
2. Given a linear program, construct its dual.
3. Explain the implications of the duality theorem.
4. Prove the weak duality theorem.
5. Prove the complementary slackness conditions.

Reading assignment

1. AA: Chapter 12, sections 12.1 and 12.3.
2. AA: Pages 1 and 15.

Exercises and review questions

• Questions on current lecture's material
  1. Give the dual of the linear program in standard form that you obtained in CLR: exercise 29.1-4.
  2. (Post your answer on the discussion board) Give feasible solutions for the primal and dual of the example above, evaluate the objective function for those, and show how they are consistent with the weak duality theorem.
  3. (Post your answer on the discussion board) Give an example of a linear program where the best integer solution has objective value greater than that for the optimal solution.

• Questions on next lecture's material
  1. Define the set cover and the vertex cover problems.
  2. Formulate set cover as an integer linear program.