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Lecture 10 Matroid

Lecture 10 Matroid. 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. Matroid. Matric Matroid. Graphic Matroid. Extension.

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Lecture 10 Matroid

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  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

  15. 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 (<=).

  16. Sufficiency

  17. A Task Scheduling Problem

  18. Unit-time Task Scheduling Input Output

  19. Independence Lemma Proof.

  20. Matroid Theorem Proof

  21. Another Example of Matroid

  22. Proof

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

  24. Puzzle

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