1. Lecture 24 Vertex Cover and Hamiltonian Cycle

2. Vertex Cover
Given a graph G=(V,E), find a minimum subset C of vertices such that every edge is incident to a vertex in C.

3. Decision Version
Given a graph G=(V,E) and positive integer k < |V|, is there a vertex cover C of size at most k?.

4. Vertex-Cover is NP-complete
Proof.

8. 2-approximation
The vertex set of a maximal matching gives 2-approximation, i.e.,
approx / opt < 2

9. Performance Ratio

10. ρ-Approximation
ρ-approximation is a polynomial-time approximation satisfying:
1 < approx(input)/opt(input) < ρ for MIN or
1 < opt(input)/approx(input) < ρ for MAX

11. Max Independent Set
An independent set is a vertex subset such that no edge exists between any two in the subset.
A vertex subset is a vertex cover if and only if its complement is an independent set.
Given a graph, find a maximum independent set.

12. Complexity and Approximation
Max Independent Set is NP-hard.
For any constant c>1, there does not exist a polynomial-time c-approximation unless NP=P.

13. Max Clique
A clique is a vertex subset which induces a complete subgraph.
A vertex subset is a clique of graph G if and only if it is an independent set of the complement of G.
:Max Clique: Given a graph, find a maximum clique.

14. Complexity and Approximation
Max Clique is NP-hard.
For any constant c>1, there does not exist a polynomial-time c-approximation unless NP=P.

15. Hamiltonian Cycle Given a graph G, does G contain a Hamiltonian cycle?
Hamiltonian cycle is a cycle passing every vertex exactly once.

16. HC is NP-complete

20. Traveling Salesman
Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.

21. HC < m TSP p
From a given graph G, we need to construct (n cities, a distance table, k).

22. Quiz Sample Is Traveling Salesman Problem NP-complete? Answer: No
Because NP contains only decision problems, TSP does not belong to NP.

23. Quiz Sample
TSP is NP-hard. Does this mean that for any A in NP, A < m TSP?
Answer: No!
It is in wide sense that if TSP can be solved in p.-t., then every problem in NP can be solved in p.-t.. p

24. Special Case
Theorem
Traveling around a minimum spanning tree is a 2-approximation.

25. Theorem
Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5-approximation

26. Minimum perfect matching on odd vertices
has weight at most 0.5 opt.