Lecture 16 Maximum Matching

# Lecture 16 Maximum 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. Alternative Path

8. Optimality Condition

9. Puzzle

10. Extension to Graph

11. Matching in Graph Maximum Matching

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

13. Alternative Path

14. Optimality Condition

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

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

17. In Each Phase

19. Max Weighted Matching

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

21. Minimum Weight Matching

22. Augmenting Path

23. Optimality Condition

24. Chinese Postman

25. 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.

26. Thanks, end.