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Discrete Optimization Lecture 3 – Part 1

Discrete Optimization Lecture 3 – Part 1. M. Pawan Kumar pawan.kumar@ecp.fr. Slides available online http:// cvn.ecp.fr /personnel/ pawan /. Questions?. Solutions for the knapsack problems?. Outline. Path Packing Description of the Algorithm Analysis of the Algorithm.

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Discrete Optimization Lecture 3 – Part 1

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  1. Discrete OptimizationLecture 3 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvn.ecp.fr/personnel/pawan/

  2. Questions? Solutions for the knapsack problems?

  3. Outline • Path Packing • Description of the Algorithm • Analysis of the Algorithm

  4. Arc Disjoint s-t Paths v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9 Set of s-t Paths with no common arcs

  5. s-t Cut v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9 t not in U C = out-arcs(U) s in U

  6. An Interesting Operation on Digraphs Find an s-t path P v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9

  7. An Interesting Operation on Digraphs Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 t v7 v9

  8. An Interesting Operation on Digraphs Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  9. Reversing vs. Removing Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  10. Reversing vs. Removing U = {v0, v4} v0 v8 s v1 v2 v3 |out-arcs(U)| decreases by 1. v4 v5 v6 t v7 v9

  11. Reversing vs. Removing U = {v0, v4} v0 v8 s v1 v2 v3 |out-arcs(U)| decreases by 2. v4 v5 v6 t v7 v9

  12. Reversing vs. Removing U = {v0} v0 v8 s v1 v2 v3 |out-arcs(U)| decreases by 1. v4 v5 v6 t v7 v9

  13. Reversing vs. Removing U = {v0} v0 v8 s v1 v2 v3 |out-arcs(U)| decreases by 1. v4 v5 v6 Unknown decrease in cut size. t v7 v9 Cannot construct an algorithm.

  14. Another Example U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 Corresponds to minimum s-t cut … of size 3 t v10

  15. Another Example U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 Corresponds to minimum s-t cut … of size 3 t v10 Why is (v2,v6) not included?

  16. Find an s-t Path U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 t v10

  17. Find an s-t Path U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 t v10

  18. Reversing vs. Removing U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 t v10

  19. Reversing vs. Removing U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 Reduces minimum s-t cut … by 1 t v10

  20. Reversing vs. Removing U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 Reduces minimum s-t cut … by 1 t v10

  21. Reversing vs. Removing U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 t v10

  22. Reversing vs. Removing U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 Reduces minimum s-t cut … by 2 t v10

  23. Reversing vs. Removing U v0 s v1 v2 v3 v4 v5 v6 v7 v8 v9 Reduces minimum s-t cut … by 2 t v10

  24. Reversing vs. Removing Removing the arcs doesn’t decrease cut size by a constant Reversing the arcs does decrease cut size by a constant A simple algorithm made possible by reversing

  25. Outline • Path Packing • Description of the Algorithm • Analysis of the Algorithm

  26. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9

  27. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 t v7 v9

  28. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  29. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  30. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  31. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  32. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 Reverse the arcs of P v4 v5 v6 For U, s in U, t not in U |out-arcs(U)| decreases by 1. t v7 v9 Size of each s-t cut decreases by 1

  33. Algorithm Find an s-t path P v0 v8 s v1 v2 v3 No more s-t paths v4 v5 v6 Size of minimum cut = 0 t v7 v9 Size of minimum cut for original digraph = 3

  34. Algorithm While there exist s-t paths Find an s-t path P Reverse the arcs of P End U is the set of vertices reachable by s in the end Minimum s-t cut = out-arcs(U)

  35. Outline • Path Packing • Description of the Algorithm • Analysis of the Algorithm

  36. Computational Complexity O(m) While there exist s-t paths Find an s-t path P O(m) Reverse the arcs of P End O(m2)

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