Introduction to Approximation Algorithms: Max 3DM and PTAS

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

7. Greedy Algorithm

8. Observation

9. Theorem

10. Max Coverage Red + Green

11. Greedy Algorithm

12. Theorem

13. Lower Bound

14. Knapsack

15. 2-approximation

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

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

18. Proof.

20. Time

21. 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).

22. Fully FTAS for Knapsack

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

26. 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

29. 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)

30. CS6382 CS7301-CS6301