Discrete Optimization: Fundamentals and Applications

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Explore the foundations of discrete optimization in MA2827 with a focus on graph theory, complexity basics, shortest path algorithms, minimum spanning trees, maximum flow, and more. Dive into concepts such as Menger's Theorem, disjoint paths, path packing, and directed graphs. Gain insights into vertex indegree and outdegree, s-t paths, and maximizing flow/minimizing cuts.


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  1. Discrete Optimization MA2827 Fondements de l optimisation discr te https://project.inria.fr/2015ma2827/ Slides courtesy of M. Pawan Kumar

  2. Recap of previous lectures Lecture 1 Graph preliminaries Complexity basics Shortest path algorithms (Dijkstra, Bellman-Ford, Floyd-Warshall) Lecture 2 Chow-Liu tree Minimum spanning tree (Prim s, Kruskal s) Maximum flow (Ford-Fulkerson, Dinits)

  3. Outline Preliminaries Menger s Theorem for Disjoint Paths Path Packing

  4. But first

  5. Find the max flow/min cut. Show the steps.

  6. Outline Preliminaries Menger s Theorem for Disjoint Paths Path Packing

  7. Directed Graphs (Digraphs) D = (V, A) v0 v1 v4 v2 v5 v3 v6 n vertices or nodes V m arcs A: ordered pairs from V

  8. Indegree of a Vertex D = (V, A) v0 v1 v4 v2 v5 v3 v6 Number of arcs entering the vertex. indeg(v0) = 1, indeg(v1) = 1, indeg(v4) = 2,

  9. Indegree of a Subset of Vertices D = (V, A) v0 v1 v4 v2 v5 v3 v6 Number of arcs entering the subset. indeg({v0,v1}) = 1, indeg({v1,v4}) = 3,

  10. Outdegree of a Vertex D = (V, A) v0 v1 v4 v2 v5 v3 v6 Number of arcs leaving the vertex. outdeg(v0) = 1, outdeg(v1) = 1, outdeg(v2) = 2,

  11. Outdegree of a Subset of Vertices D = (V, A) v0 v1 v4 v2 v5 v3 v6 Number of arcs leaving the subset. outdeg({v0,v1}) = 1, outdeg({v1,v4}) = 2,

  12. s-t Path D = (V, A) v0 v1 v4 v2 v5 v3 v6 Sequence P = (s=v0,a1,v1, ,ak,t=vk), ai = (vi-1,vi) Vertices s=v0,v1, ,t=vk are distinct

  13. S-T Path S D = (V, A) v0 T v1 v4 v2 v5 v3 v6 S and T are subsets of V Any st-path where s S and t T

  14. S-T Path S D = (V, A) v0 T v1 v4 v2 v5 v3 v6 S and T are subsets of V Any st-path where s S and t T

  15. Outline Preliminaries Menger s Theorem for Disjoint Paths Path Packing

  16. Vertex Disjoint S-T Paths S v0 v8 v1 v2 v3 v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

  17. Vertex Disjoint S-T Paths S v0 v8 v1 v2 v3 v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

  18. Vertex Disjoint S-T Paths S v0 v8 Common Vertex v7 v1 v2 v3 v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

  19. Vertex Disjoint S-T Paths S v0 v8 Common Vertex v0 v1 v2 v3 v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

  20. Internally Vertex Disjoint s-t Paths s v0 v8 v1 v2 v3 v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

  21. Internally Vertex Disjoint s-t Paths s v0 v8 v1 v2 v3 v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

  22. Internally Vertex Disjoint s-t Paths s v0 v8 Common Vertex v5 v1 v2 v3 v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

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

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

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

  26. Outline Preliminaries Menger s Theorem for Disjoint Paths Vertex Disjoint S-T Paths Internally Vertex Disjoint s-t Paths Arc Disjoint s-t Paths Path Packing

  27. Vertex Disjoint S-T Paths S v0 v8 Maximum number of disjoint paths? v1 v2 v3 Minimum size of S-T disconnecting vertex set !! v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

  28. S-T Disconnecting Vertex Set S v0 v8 v1 v2 v3 v4 v5 v6 T v7 v9 Subset U of V which intersects with all S-T Paths

  29. S-T Disconnecting Vertex Set S v0 v8 v1 v2 v3 v4 v5 v6 T v7 v9 Subset U of V which intersects with all S-T Paths

  30. S-T Disconnecting Vertex Set S v0 v8 v1 v2 v3 v4 v5 v6 T v7 v9 Subset U of V which intersects with all S-T Paths

  31. S-T Disconnecting Vertex Set S v0 v8 v1 v2 v3 v4 v5 v6 T v7 v9 Subset U of V which intersects with all S-T Paths

  32. Connection S v0 v8 Maximum number of disjoint paths v1 v2 v3 Minimum size of S-T disconnecting vertex set !! v4 v5 v6 T v7 v9

  33. Mengers Theorem S v0 v8 Maximum number of disjoint paths v1 v2 v3 = Minimum size of S-T disconnecting vertex set !! v4 v5 v6 T Proof ? v7 v9 Mathematical Induction on |A|

  34. Mengers Theorem S v0 v8 True for |A| = 0 v1 v2 v3 Assume it is true for |A| < m v4 v5 v6 To be continued (Try working out the rest) T v7 v9 Mathematical Induction on |A|

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