1. Lecture 10 Matroid

2. Independent System Consider a finite set S and a collection C of subsets of S. (S,C) is called an independent system if i.e., it is hereditary. Each subset in C is called an independent set.

3. Matroid

4. Matric Matroid

5. Graphic Matroid

6. Extension

7. Maximal Independent Set Theorem

8. Proof

9. Basis

10. Weighted Independent System

11. Minimum Spanning Tree

12. Greedy Algorithm MAX

13. Theorem

14. Proof

17. About Matriod Theorem An independent system (S,C) is a matroid iff for any cost function c( ), the greedy algorithm MAX gives a maximum solution. Proof. (=>) Next, we show (<=).

18. Sufficiency

19. A Task Scheduling Problem

20. Unit-time Task Scheduling Input Output

21. Independence Lemma Proof.

22. Matroid Theorem Proof

23. K=?

24. Another Example of Matroid

25. Proof

26. What we learnt in this lecture? What is matroid?. matric matroid and graphic matroid. Relationship between matroid and greedy algorithm.

27. Puzzle