Lecture 2

Learning objectives

After this class, you should be able to:

1. Define the optimization version of vertex cover, and, in particular, cardinality vertex cover.
2. Give a lower bound on the size of a vertex cover, based on matching.
3. Describe the approximation algorithm for vertex cover based on matching, prove that its solution is feasible, and that it is factor 2.
4. Show that the bound on the approximation factor given above is tight.
5. Show that the size of a maximal matching may be as small as half the size of the vertex cover. (Consequently, it is unlikely that we can develop another algorithm based on maximal matching, which will have a smaller approximation factor.)
6. Derive lower bounds and approximation algorithms for similar problems.

Reading assignment

1. CLR: Chapter 35, up to (and including) sec 35.1; AA: Chapter 1, up to (and including) section 1.1.
2. AA: Chapter 2, pages 15 and 17.

Exercises and review questions

• Questions on current lecture's material
  1. Prove the correctness of the sizes given in AA example 1.5 for (i) maximal matching and (ii) optimal vertex cover.

     Example 1.5: The lower bound, of size of a maximal matching, is half the size of an optimal vertex cover for the following infinite family of instances. Consider the complete graph Kn, where n is odd. The size of any maximal matching is (n-1)/2, whereas the size of an optimal cover is n-1.

  2. AA: Exercise 1.1: Give a factor 1/2 algorithms for the following. (Acyclic subgraph) Given a directed graph G = (V,E), pick a maximum cardinality set of edges from E so that the resulting subgraph is acyclic. [Hint: Arbitrarily number the vertices and pick the bigger of the two sets, the forward-going edges and the backward-going edges. What scheme are you using to upper bound OPT?]

  3. (Post your answer on the discussion board) CLR: page 1026 gives the time complexity for the vertex cover approximation algorithm is given as O(V+E). Justify this claim.

• Questions on next lecture's material
  1. Give an example of the set cover problem, and give an optimal solution for it.
  2. Prove that Hn is O(log n).