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

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6. Note: Every edge has capacity 1.

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

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

17. Alternative Path

18. Optimality Condition

20. Puzzle

21. Extension to Graph

22. Matching in Graph Maximum Matching

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

24. Alternative Path

25. Optimality Condition

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

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

28. In Each Phase

29. Running Time Reading Material

30. Max Weighted Matching

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

32. Minimum Weight Matching

33. Augmenting Path

34. Optimality Condition

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45. Chinese Postman

47. 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.