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

5. 1 1

6. Note: Every edge has capacity 1.

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

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.

28. In Each Phase

30. Running Time
Reading Material

31. Max Weighted Matching

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

33. Minimum Weight Matching

34. Augmenting Path

35. Optimality Condition

46. Chinese Postman

48. 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.

49. Thanks, end.