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Exploration of Minimum Spanning Trees and Greedy Strategies in Graph Theory

This lecture delves into the concepts of minimum spanning trees (MST) and greedy algorithms, focusing on their properties and applications in optimization. It elaborates on the maximum weight subset of non-conflicting jobs scheduled on a single processor, emphasizing the importance of independent sets in interval graphs. Additional discussions include effective profit functions, the MaxIS optimization algorithm, and the relationship between matroids and independent systems. Illustrative examples and key points underscore the principles of self-reducibility and exchange properties related to these topics.

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Exploration of Minimum Spanning Trees and Greedy Strategies in Graph Theory

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Presentation Transcript

  1. Lecture 14 Overview Minimum Spanning Tree and Greedy Strategy

  2. A Property of MST

  3. Local Ratio Method

  4. 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.

  5. Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt 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

  6. Maximal Solutions • We say that a feasible schedule is I-maximalif 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

  7. Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt 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 Îbe an interval that ends first;

  8. Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt 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.

  9. Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt Independent Set in Interval Graphs:An Optimization Algorithm • Algorithm MaxIS( S, p ) • If S = Φ then returnΦ ; • If ISp(I) 0 then returnMaxIS( 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 returnIS else return IS {Î};

  10. Slide from http://www.cs.technion.ac.il/~reuven/STOC2000.ppt Running Example P(I5) = 3 -4 P(I6) = 6 -4 -2 P(I3) = 5 -5 P(I2) = 3 -5 P(I1) = 5 -5 P(I4) = 9 -5 -4 -4 -5 -2

  11. Overview

  12. RevisitMinimum Spanning Tree

  13. Exchange Property

  14. Self-Reducibility

  15. Max Independent Set in Matroid

  16. Exchange Property

  17. Self-Reducibility

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

  19. Minimum Weight Arborescence

  20. Definition

  21. Problem

  22. Key Point 1

  23. Key Point 2

  24. Why?

  25. Key Point 3 0

  26. Relationship betweenMatroid and Independent System

  27. An Example of Matroid

  28. Proof

  29. Theorem Every independent system is an intersection of several matroids.

  30. circuit • A minimal dependent set is called a circuit. • Let A1, …, Ak be all circuits of independent system (E,C).

  31. Theorem If independent system (S,C) is the intersection of k matroids (S,Ci), then for any subset F of S, u(F)/v(F) <k.

  32. Proof

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