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.

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Presentation transcript:

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

Maximal Independent Set Theorem

Proof

Basis

Weighted Independent System

Minimum Spanning Tree

Greedy Algorithm MAX

Theorem

Proof

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 (<=).

Sufficiency

A Task Scheduling Problem

Unit-time Task Scheduling Input Output

Independence Lemma Proof.

Matroid Theorem Proof

K=?

Another Example of Matroid

Proof

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

Puzzle