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Lecture 21 More Approximation Algorithms

Lecture 21 More Approximation Algorithms. Introduction. Maximum 3DM. 3-Approximation. Any maximal 3DM is a 3-approximation for max 3DM. This is because in the maximum 3DM, every edge (3-set) must have at least one vertex covered by the maximal 3DM. Min Set Cover. Red + Green.

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Lecture 21 More Approximation Algorithms

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  1. Lecture 21MoreApproximation Algorithms Introduction

  2. Maximum 3DM

  3. 3-Approximation • Any maximal 3DM is a 3-approximation for max 3DM. • This is because in the maximum 3DM, every edge (3-set) must have at least one vertex covered by the maximal 3DM.

  4. Min Set Cover Red + Green

  5. Greedy Algorithm

  6. Observation

  7. Theorem

  8. Max Coverage Red + Green

  9. Greedy Algorithm

  10. Theorem

  11. Lower Bound

  12. Knapsack

  13. 2-approximation

  14. PTAS • A problem has a PTAS (polynomial-time approximation scheme) if for any ε > 0, it has a (1+ε)-approximation.

  15. Knapsack has PTAS • Classify: for i < m, ci< a= cG, for i > m+1, ci > a. • Sort • For

  16. Proof.

  17. Time

  18. Fully PTAS • A problem has a fully PTAS if for any ε>0, it has (1+ε)-approximation running in time poly(n,1/ε).

  19. Fully FTAS for Knapsack

  20. Pseudo Polynomial-time Algorithm for Knapsak • Initially,

  21. Time • outside loop: O(n) • Inside loop: O(nM) where M=max ci • Core: O(n log (MS)) • Total O(n M log (MS)) • Since input size is O(n log (MS)), this is a pseudo-polynomial-time due to M=2 3 log M

  22. Complexity of Approximation • FPTAS (e.g., Knapsack) • PTAS (e.g., Knapsack) • Constant-approximation (e.g., vertex-cover) • -approximation (e.g., set cover) • -approximation (e.g., max clique)

  23. CS6382 CS7301-CS6301

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