1. Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du

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

3. Definition

4. Proof: Given a graph G=(V,E), define a distance table on V as follows: Theorem

5. Contradiction Argument Suppose r-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow: r-approximation solution < r |V| if and only if G has a Hamiltonian cycle

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

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

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

9. Knapsack

10. Definition

11. Theorem Proof.

12. Theorem

13. Classify: for i < m, c i < a= c G, for i > m+1, c i > a. Sort For Algorithm

14. Proof.

16. Time

17. MAX3SAT

18. Theorem

19. This an important result proved using PCP system. Theorem

20. Class MAX SNP (APX?)

21. L-reduction

23. VC-b Theorem

24. 12 3 4 5 12 3 4 5 GG’ v

27. Properties (P1) (P2)

30. PTAS MAX SNP

35. MAX SNP-complete (APX-complete) Theorem

36. MAX3SAT-3 Theorem

37. VC-4 is MAX SNP-complete Proof.

39. Theorem Proof.

40. Theorem Proved using PCP system

41. Theorem MCDS

43. CLIQUE Theorem Proved with PCP system.

44. 1 2 Exercises

45. 3

46. hint

47. Min-2-DS is MAX SNP-complete in the case that all given pools have size at most 2. 4 Prove that

48. 5. Is TSP with triangular inequality MAX SNP-complete?