1 / 26

Understanding Matroids and Independent Systems: Principles and Theorems

This lecture explores the concept of matroids and independent systems, defining these structures through finite sets and collections of subsets. We delve into hereditary properties and the role of independent sets. Key topics include different types of matroids, the definition and implications of maximal independent sets, and proofs of fundamental theorems such as the Greedy Algorithm for optimization problems. We also discuss practical applications like task scheduling and analyze the relationships between matroids and greedy algorithms, enhancing our understanding of mathematical structures in combinatorial optimization.

sonja
Télécharger la présentation

Understanding Matroids and Independent Systems: Principles and Theorems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related