1. Lecture 21 More Approximation 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,
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