Chapter 10 Complexity of Approximation (1) L-Reduction

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

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 Theorem • Traveling around a minimum spanning tree is a 2-approximation.

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

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

9. Knapsack

10. Definition

11. Theorem Proof.

12. Theorem

13. Algorithm • Classify: for i < m, ci< a= cG, for i > m+1, ci > a. • Sort • For

14. Proof.

16. Time

17. MAX3SAT

18. Theorem

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

20. ClassMAX SNP (APX?)

21. L-reduction

23. VC-b Theorem

24. 1 2 1 2 v 3 3 4 5 4 5 G G’

27. Properties (P1) (P2)

30. MAX SNP PTAS

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

36. MAX3SAT-3 Theorem

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

39. Theorem Proof.

40. Theorem Theorem Proved using PCP system

41. MCDS Theorem

43. CLIQUE Theorem Proved withPCP system.

44. Exercises 1 2

45. 3

46. hint

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

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