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Chapter 10 Complexity of Approximation (1) L-Reduction

Chapter 10 Complexity of Approximation (1) L-Reduction. Ding-Zhu Du. Traveling Salesman. Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once. Definition. Theorem. Proof: Given a graph G=(V,E), define a distance table on V as follows:.

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Chapter 10 Complexity of Approximation (1) L-Reduction

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  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.

  15. Time

  16. MAX3SAT

  17. Theorem

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

  19. ClassMAX SNP (APX?)

  20. L-reduction

  21. VC-b Theorem

  22. 1 2 1 2 v 3 3 4 5 4 5 G G’

  23. Properties (P1) (P2)

  24. MAX SNP PTAS

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

  26. MAX3SAT-3 Theorem

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

  28. Theorem Proof.

  29. Theorem Theorem Proved using PCP system

  30. MCDS Theorem

  31. CLIQUE Theorem Proved withPCP system.

  32. Exercises 1 2

  33. 3

  34. hint

  35. 4 Prove that • Min-2-DS is MAX SNP-complete in the case that all given pools have size at most 2.

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

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