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Lecture 21 Primal-Dual in Algorithms

Lecture 21 Primal-Dual in Algorithms. Primal Type. At each iteration, a feasible solution is updated to approach the optimal. Dual Type. At each iteration, a non-feasible “solution” is modified to approach to the feasibility.

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Lecture 21 Primal-Dual in Algorithms

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  1. Lecture 21Primal-Dual in Algorithms

  2. Primal Type • At each iteration, a feasible solution is updated to approach the optimal.

  3. Dual Type • At each iteration, a non-feasible “solution” is modified to approach to the feasibility. • E.g., Kruskal Algorithm and Prim Algorithm for minimum spanning tree.

  4. Primal-Type Algorithm for Minimum Spanning Tree

  5. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Example 4 2 6 1 5 3 7

  6. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Find a Cycle 4 2 6 1 5 3 7

  7. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Delete the longest edge 4 2 6 1 5 3 7

  8. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Find a cycle 4 2 6 1 5 3 7

  9. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Delete a longest edge 4 2 6 1 5 3 7

  10. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Find a cycle 4 2 6 1 5 3 7

  11. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Delete a longest edge 4 2 6 1 5 3 7

  12. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Find a cycle 4 2 6 1 5 3 7

  13. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Delete a longest edge 4 2 6 1 5 3 7

  14. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Find a cycle 4 2 6 1 5 3 7

  15. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Delete a longest edge 4 2 6 1 5 3 7

  16. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Find a cycle 4 2 6 1 5 3 7

  17. 10 8 2 4 6 35 15 1 25 20 30 17 21 40 3 5 7 15 11 Delete a longest edge 4 2 6 1 5 3 7

  18. Maximum Flow Primal-type • Ford-Fulkerson algorithm • Hopcroft–Karp algorithm Dual-type • Push-relabel algorithm Goldberg-Tarjan algorithm

  19. Transportation Problem

  20. Primal-Type Algorithm for Chinese Postman Problem

  21. Primal-Dual Method for LP

  22. Primal-Dual Type

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