The document discusses bipartite matching and describes how to find the maximum matching between two disjoint sets using the augmenting path algorithm. It explains that the algorithm works by searching for augmenting paths between unmatched vertices, reversing any that are found to increase the size of the matching, and repeating until no more augmenting paths exist. The time complexity of this algorithm is O(V×E) where V is the number of vertices and E is the number of edges.

1. Bipartite Matching
郭至軒（KuoE0）
KuoE0.tw@gmail.com
KuoE0.ch

2. Bipartite Graph
two disjoint sets
every edge connects between them

3. Matching
A subset of edges, no two of which share an
endpoint.

4. Valid Matching
not matching edge
matching edge
bachelor

5. Invalid Matching
not matching edge
matching edge
shared

6. Bipartite Matching
objective: more matching

7. 1 5
2 6
3 7
4 8
not matching edge 9
matching edge

8. Alternating Path
A path in which the edges belong alternatively to the
matching and not to the matching.

9. [1 - 5 - 2 - 9] is an alternating path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

10. [1 - 5 - 2 - 9] is an alternating path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

11. [1 - 5 - 2 - 6 - 3] is not an alternating path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

12. [1 - 5 - 2 - 6 - 3] is not an alternating path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

13. Alternating Cycle
A cycle in which the edges belong alternatively to
the matching and not to the matching.

14. [1 - 5 - 2 - 9 - 1] is an alternating cycle.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

15. [1 - 5 - 2 - 9 - 1] is an alternating cycle.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

16. Reverse the alternating cycle, cardinality won’t change.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

17. Maximum Matching
Augmenting Path Algorithm

18. Augmenting Path
An alternating path that starts from and ends on
unmatched vertices.

19. [2 - 5 - 1 - 9] is an augmenting path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

20. [2 - 5 - 1 - 9] is an augmenting path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

21. Reverse the augmenting path, cardinality will increase.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge

22. Augmenting Path
• The length of an augmenting path
always is odd.
• Reversing an augmenting path will
increase the cardinality.
• If there is no augmenting path, the
cardinality is maximum.

23. Algorithm
1. Try to build augmenting paths from all
vertices in one side.
2. Travel on graph.
3. If an augmenting path exists, reverse
the augmenting path to increase
cardinality.
4. If no augmenting path exists, ignore
this vertex.
5. Repeat above step until there no
augmenting path exists.

24. [1 - 5] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 0
9
not matching edge
matching edge

25. [1 - 5] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 0
9
not matching edge
matching edge

26. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 1
9
not matching edge
matching edge

27. [2 - 5 - 1- 9] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 1
9
not matching edge
matching edge

28. [2 - 5 - 1- 9] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 1
9
not matching edge
matching edge

29. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 2
9
not matching edge
matching edge

30. [3 - 6] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 2
9
not matching edge
matching edge

31. [3 - 6] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 2
9
not matching edge
matching edge

32. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 3
9
not matching edge
matching edge

33. [4 - 5 - 2 - 6 - 3 - 7] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 3
9
not matching edge
matching edge

34. [4 - 5 - 2 - 6 - 3 - 7] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 3
9
not matching edge
matching edge

35. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 4
9
not matching edge
matching edge

36. Max cardinality is 4.
1 5
2 6
3 7
4 8
9
not matching edge
matching edge

37. Time Complexity
O(V × E)
V: number of vertices
E: number of edges

38. Source Code
// find an augmenting path
bool find_aug_path( int x ) {
for ( int i = 0; i < ( int )vertex[ x ].size(); ++i ) {
int next = vertex[ x ][ i ];
// not in augmenting path
if ( !visit[ next ] ) {
// setup this vertex in augmenting path
visit[ next ] = 1;
/*
* If this vertex is a unmatched vertex, reverse
augmenting path and return.
* If this vector is a matched vertex, try to reverse
augmenting path and continue find an unmatched vertex.
*/
if ( lnk2[ next ] == -1 || find_aug_path( lnk2[ next ] ) )
{
lnk1[ x ] = next, lnk2[ next ] = x;
return 1;
}
}
}
return 0;
}

39. Practice Now
UVa 670 - The dog task

40. Maximum Weight Matching
Hungarian Algorithm
(Kuhn-Munkres Algorithm)

41. 5
1 4
2 3
1
2 5
3 5 -2 6

42. weight adjustment
If add (subtract) some value on all edges connected with
vertex X, the maximum matching won’t be effected.
1 4 1 4
a a+d
b b+d
2 5 2 5
c c+d
3 6 3 6
matching edge not matching edge

43. vertex labeling
For convenient, add a variable on vertices to denote the
value add (subtract) on edges connected with them.
1 4 1 4
a a+d
b b+d
d 2
c
5 ≡ 2
c+d
5
3 6 3 6
matching edge not matching edge

44. vertex labeling
l(x)+l(y)≥w(x,y), for each edge(x,y)
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0

45. minimize vertex labeling
Admissible Edge: l(x)+l(y)=w(x,y)
5
2 1 4 3
2 3
1
0 2 5 1
4 3 5 -2 6 0

46. convert
maximum weight problem
to
minimum vertex labeling problem

47. Augmenting Path with Admissible Edge
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
[1 - 4] is an augment path with admissible edges.

48. Augmenting Path with Admissible Edge
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
[1 - 4] is an augment path with admissible edges.

49. Augmenting Path with Admissible Edge
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
Reverse it!
Current Weight = 5

50. Augmenting Path with Admissible Edge
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
No augmenting path found start from vertex 2.

51. Augmenting Path with Admissible Edge
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
No augmenting path found start from vertex 2.

52. Adjust Vertex Labeling
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
relax value d = min(l(x)+l(y)-w(x,y))
x: vertex in alternating path
y: vertex y not in alternating path

53. Adjust Vertex Labeling
5
5 1 4 0
2 3
1
3 2 5 0
5 3 5 -2 6 0
R(1,6)=3
R(2,5)=2 relax d = 2
R(2,6)=5

54. Adjust Vertex Labeling
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
l(x) subtract value d.
l(y) add value d.

55. Continue to Find Augmenting Path
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
[2 - 5] is an augment path with admissible edges.

56. Continue to Find Augmenting Path
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
[2 - 5] is an augment path with admissible edges.

57. Continue to Find Augmenting Path
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
Reverse it!
Current Weight = 6

58. Continue to Find Augmenting Path
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
No augmenting path found start from vertex 3.

59. Continue to Find Augmenting Path
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
No augmenting path found start from vertex 3.

60. Adjust Vertex Labeling
5
3 1 4 2
2 3
1
1 2 5 0
5 3 5 -2 6 0
R(1,6)=1
relax d = 1
R(2,6)=3

61. Adjust Vertex Labeling
5
2 1 4 3
2 3
1
0 2 5 1
4 3 5 -2 6 0
l(x) subtract value d.
l(y) add value d.

62. Continue to Find Augmenting Path
5
2 1 4 3
2 3
1
0 2 5 1
4 3 5 -2 6 0
[3 - 5 - 2 - 4 - 1 - 6] is an augment path with
admissible edges.

63. Continue to Find Augmenting Path
5
2 1 4 3
2 3
1
0 2 5 1
4 3 5 -2 6 0
[3 - 5 - 2 - 4 - 1 - 6] is an augment path with
admissible edges.

64. Continue to Find Augmenting Path
5
2 1 4 3
2 3
1
0 2 5 1
4 3 5 -2 6 0
Reverse it!
Current Weight = 10

65. Maximum Weight Matching = 10
5
2 1 4 3
2 3
1
0 2 5 1
4 3 5 -2 6 0

66. Algorithm
1. Initial vertex labeling to fit l(x)+l(y)≥w(x,y)
2. Find augmenting paths composed with
admissible edges from all vertices in one side.
3. If no augmenting path exists, adjust vertex
labeling until the augmenting path found.
4. If augmenting path exists, continue to find next
augmenting path.
5. Repeat above step until there no augmenting
path exists.
6. Calculate the sum of weight on matching edges.

67. Practice Now
POJ 2195 - Going Home

68. Problem List
UVa 670
UVa 753
UVa 10080
UVa 10092
UVa 10243
UVa 10418
UVa 10984
POJ 3565

69. Reference
• http://en.wikipedia.org/wiki/Bipartite_graph
• http://en.wikipedia.org/wiki/Matching_(graph_theory)
• http://www.flickr.com/photos/marceau_r/5244129689
• http://www.sanfilippo-chianti.it/offerta-svalentino.html
• http://www.csie.ntnu.edu.tw/~u91029/Matching.html

70. Thank You for Your
Listening.