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Lecture 16 Maximum Matching

This lecture discusses the incremental method for achieving maximum matching in bipartite graphs, highlighting the transformation from one feasible solution to another to optimize the objective function. It examines the effectiveness of augmenting paths in bipartite graphs, introduces the Hopcroft-Karp algorithm as an adaptation for maximum flow problems, and delves into concepts like maximum and minimum weight matchings. The lecture also clarifies the complexities and conditions required for transformations between matching types, specifically relating to the Chinese Postman problem and minimum weight perfect matching.

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Lecture 16 Maximum Matching

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  1. Lecture 16 Maximum Matching

  2. Incremental Method • Transform from a feasible solution to another feasible solution to increase (or decrease) the value of objective function.

  3. Matching in Bipartite Graph Maximum Matching

  4. 1 1

  5. Note: Every edge has capacity 1.

  6. 1. Can we do augmentation directly in bipartite graph? 2. Can we do those augmentation in the same time?

  7. 1. Can we do augmentation directly in bipartite graph? Yes!!!

  8. Alternative Path

  9. Optimality Condition

  10. Puzzle

  11. Extension to Graph

  12. Matching in Graph Maximum Matching

  13. Note • We cannot transform Maximum Matching in Graph into a maximum flow problem. • However, we can solve it with augmenting path method.

  14. Alternative Path

  15. Optimality Condition

  16. 2. Can we do those augmentation in the same time?

  17. Hopcroft–Karp algorithm • The Hopcroft–Karp algorithm may therefore be seen as an adaptation of the Edmonds-Karp algorithm for maximum flow.

  18. In Each Phase

  19. Running Time Reading Material

  20. Max Weighted Matching

  21. Maximum Weight Matching It is hard to be transformed to maximum flow!!!

  22. Minimum Weight Matching

  23. Augmenting Path

  24. Optimality Condition

  25. Chinese Postman

  26. Minimum Weight Perfect Matching • Minimum Weight Perfect Matching can be transformed to Maximum Weight Matching. • Chinese Postman Problem is equivalent to Minimum Weight Perfect Matching in graph on odd nodes.

  27. Thanks, end.

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