Maximum matching in bipartite graphs iterative improvement method

1. Maximum Matching in Bipartite
Graphs
Iterative Improvement Method
Dr. P. Subathra
Prof/ IT
KAMARAJ College of Engg. & Tech
(AUTONOMOUS)
Madurai
Tamil Nadu
India

2. Maximum Matching in Bipartite Graphs
• The Maximum Matching in the Bipartite Graph
can be done through Iterative Improvement
methodology
• Two different approaches are available:
1. Ford Fulkerson’s Method
2. Shortest Augmenting Path Method
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
2

3. 1. Ford Fulkerson’s Augmenting
Path Method
3
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

4. Ford Fulkerson’s Augmenting Path
Method
4
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

5. Ford Fulkerson’s Augmenting Path
Method
5
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

6. Ford Fulkerson’s Augmenting Path
Method
2,6,1,7
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

7. Ford Fulkerson’s Augmenting Path
Method
2,6,1,7
7
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

8. Ford Fulkerson’s Augmenting Path
Method
2,6,1,7
8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

9. Ford Fulkerson’s Augmenting Path
Method
2,6,1,7
9
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

10. Ford Fulkerson’s Augmenting Path
Method
2,6,1,7
10
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

11. Ford Fulkerson’s Augmenting Path
Method
3,6,1  No path
11
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

12. Ford Fulkerson’s Augmenting Path
Method
3,6,1  No path
X
12
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

13. Ford Fulkerson’s Augmenting Path
Method
3,6,2  No path
13
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

14. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
14
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

15. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
15
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

16. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
16
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

17. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
17
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

18. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
18
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

19. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
19
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

20. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
20
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

21. Ford Fulkerson’s Augmenting Path
Method
3,8,4,9,5,10
21
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

22. Ford Fulkerson’s Augmenting Path
Method
22
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

23. Maximum Matching Theorem
23
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

24. 2. Shortest Augmenting Path - BFS
24
Initial Matching Pairs = 2
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

25. Shortest Augmenting Path - BFS
25
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

26. Shortest Augmenting Path - BFS
1
26
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

27. Shortest Augmenting Path - BFS
27
Number of Matching Pairs = 3
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

28. Shortest Augmenting Path - BFS
28
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

29. Shortest Augmenting Path - BFS
2
29
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

30. Shortest Augmenting Path - BFS
2 3
30
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

31. Shortest Augmenting Path - BFS
6
31
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

32. Shortest Augmenting Path - BFS
8
32
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

33. Shortest Augmenting Path - BFS
7
1
33
6 8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

34. Shortest Augmenting Path - BFS
7
1
34
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

35. Shortest Augmenting Path - BFS
7
1
35
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

36. Shortest Augmenting Path - BFS
7
1
36
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

37. Shortest Augmenting Path - BFS
7
1
37
Number of Matching Pairs = 4
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

38. Shortest Augmenting Path - BFS
38
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

39. Shortest Augmenting Path - BFS
39
3
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

40. Shortest Augmenting Path - BFS
40
6
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

41. Shortest Augmenting Path - BFS
41
6 8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

42. Shortest Augmenting Path - BFS
42
6 8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

43. Shortest Augmenting Path - BFS
43
10
6 8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

44. Shortest Augmenting Path - BFS
44
10
6 8
Augmenting Path From Node 10 : 3-84-10
(Trace backward from 10 as 10-4-8-3)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

45. Shortest Augmenting Path - BFS
45
10
6 8
Augmenting Path From Node 10 : 3-84-10
(Trace backward from 10 as 10-4-8-3)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

46. Shortest Augmenting Path - BFS
46
10
6 8
Augmenting Path From Node 10 : 3-84-10
(Trace backward from 10 as 10-4-8-3)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

47. Shortest Augmenting Path - BFS
47
10
Augmenting Path From Node 10 : 3-84-10
(Trace backward from 10 as 10-4-8-3)
8
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

48. Shortest Augmenting Path - BFS
48
10
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

49. Shortest Augmenting Path - BFS
49
10
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

50. Shortest Augmenting Path - BFS
50
10
Total Number of Matching Pairs = 5
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

51. Shortest Augmenting Path - BFS
51
10
Queue : Empty
Maximum Matching Achieved
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India

52. 52
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India