1. Lecture 11 Overview
Self-Reducibility

2. Overview on Greedy Algorithms

3. Revisit Minimum Spanning Tree

4. Exchange Property

5. Self-Reducibility

6. Max Independent Set in Matroid

7. Exchange Property

8. Self-Reducibility

10. Overview on Greedy Algorithms
Self-Reducibility
Exchange Property
Matroid

11. Local Ratio Method

12. Basic Idea
Proof

13. Basic Idea

14. Minimum Spanning Tree

15. Activity Selection

16. Puzzle

17. Independent Set in Interval Graphs
Activity 9
Activity 8
Activity 7
Activity 6
Activity 5
Activity 4
Activity 3
Activity 2
Activity 1
time
We must schedule jobs on a single processor with no preemption.
Each job may be scheduled in one interval only.
The problem is to select a maximum weight subset of non-conflicting jobs.

18. Independent Set in Interval Graphs
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Independent Set in Interval Graphs
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity1
time
Maximize
s.t.
For each instance I
For each time t

19. Maximal Solutions
We say that a feasible schedule is I-maximal if either it contains instance I, or it does not contain I but adding I to it will render it infeasible.
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity1 I2 I1
time
The schedule above is I1-maximal and also I2-maximal

20. An effective profit function
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An effective profit function
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity1
P1=0
P1= P(Î)
P1=0
P1=0
P1=0
P1= P(Î)
P1=0
P1= P(Î)
P1= P(Î) Î
Let Î be an interval that ends first;

21. An effective profit function
Slide from
An effective profit function
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity1
P1=0
P1= P(Î)
P1=0
P1=0
P1=0
P1= P(Î)
P1=0
P1= P(Î)
P1= P(Î) Î
For every feasible solution x: p1 ·x  p(Î)
For every Î-maximal solution x: p1 ·x  p(Î)
Every Î-maximal is optimal.

22. Independent Set in Interval Graphs: An Optimization Algorithm
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Independent Set in Interval Graphs: An Optimization Algorithm
Algorithm MaxIS( S, p )
If S = Φ then return Φ ;
If I  S p(I) 0 then return MaxIS( S - {I}, p);
Let Î  S that ends first;
I  S define: p1 (I) = p(Î)  (I in conflict with Î) ;
IS = MaxIS( S, p- p1 ) ;
If IS is Î-maximal then return IS else return IS  {Î};

23. Running Example P(I5) = 3 -4 P(I6) = 6 -4 -2 P(I3) = 5 -5 P(I2) = 3 -5
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Running Example
P(I5) =
P(I6) =
P(I3) =
P(I2) =
P(I1) =
P(I4) = -4 -5 -2

24. Solution 1
Solution 2

25. Minimum Weight Arborescence

26. Definition

27. Problem

28. Key Point 1

29. Key Point 2

30. Why?

31. Key Point 3

32. Matroid
Greedy
Local Ratio
Divide-and-Conquer
Dynamic Programming
Self-reducibility

33. A Property of MST