Lecture 10

Learning objectives

After this class, you should be able to:

1. Given a set cover instance, and a solution to its relaxed LP formulation, give an approximation to an optimal set cover, using AA:Algorithm 14.1.
2. Prove that algorithm 14.1 yields a feasible solution, and derive its approximation factor.
3. Show how LP can be used to derive a factor 2 approximation algorithm for vertex cover.

Reading assignment

1. Chapter 14, section 14.1.
2. Section 12.1.

Exercises and review questions

• Questions on current lecture's material
  1. Consider the following alternative rounding scheme to a solution from the relaxed LP formulation for set cover. If the value for a set is > 0.5, we round it to 1, otherwise we round it to 0. Will this scheme yield an approximate solution with an approximation factor of 2? Justify your answer. (You will need to consider two factors: (i) Is a "solution" obtained feasible, and (ii) does it satisfy the approximation factor.)
  2. Explain example 14.3. Note that a hypergraph is a generalization of a graph, where a hyper-edge is a set of vertices (unlike in a graph, where it is a set of exactly two vertices).

• Questions on next lecture's material
  1. Give the dual of the relaxed version of the set cover linear program, and give the complementary slackness conditions for it.