1. 1 1
Find a maximum Matching ? 2 2 3 3 4 4 5 5
Women
Men

2. 1 1
Find a maximum matching by eyes. 2 2 3 3 4 4 5 5
Women
Men

3. Find a maximum matching by computer.1 1
Find a maximum matching by computer. 2 2
Restrictions:
Find Max 3 3 4 4 5 5
Women
Men

4. Not a good way!! 1 1 Find a maximum matching by computer. 2 2
Restriction: x12+x13=1, x23=1, x33+x34+x35=1, x41+x42+x44=1, x55=1, x41=1, x12+x42=1, x13+x23+x31=1, x34+x44=1, x35+x55=1, xij=0 or 1.
Find Max 3 3 4 4 5 5
Not a good way!!
Women
Men

5. 1 1
Find a maximum Matching by computer again? 2 2 3 3
This is a maximal matching, but not a maximum matching. 4 4 5 5
Women
Men

6. 1 1
Find a maximum Matching by computer again? 2 2 3
1. Find an augmentative path. 3 4 4 5 5
Women
Men

7. 1 1
Find a maximum Matching by computer again? 2 2 3
1. Find an augmentative path.
2. Add the edges in the path, but not in matching. 3 4 4 5 5
Women
Men

8. 1 1
Find a maximum Matching by computer again? 2 2 3
1. Find an augmentative path.
2. Add the edges in the path, but not in matching.
3. Delete the edges in the path and in the matching. 3 4 4 5 5
Women
Men

9. 1 1
Find a maximum Matching by computer again? 2 2 3
Now we have 4 pairs 3 4 4 5 5
Women
Men

10. 1 1
Find a maximum Matching by computer again? 2 2
1. Keep doing this procedure. 3 3 4 4 5 5
Women
Men

11. 1 1
Find a maximum Matching by computer again? 2 2
1. Keep doing this procedure. 3 3 4 4
Wrong!! This is not an augmentative path. 5 5
Women
Men

12. 1 1
Find a maximum Matching by computer again? 2 2
1. Keep doing this procedure. 3 3 4 4 5 5
Women
Men

13. 1 1
Find a maximum Matching by computer again? 2 2
1. Keep doing this procedure. 3 3 4 4 5 5
Women
Men

14. 1 1
Find a maximum Matching by computer again? 2 2
Keep doing this procedure.
When there exists no augmentative path, we find a maximum matching. 3 3 4 4 5 5
Women
Men

15. 3 1 1 1 5 3 2 3 4 2 1 2 3 5
Find maximum water flow with given capacities in the one-way pipes.

16. 3/3
1/1
1/1
1/1
5/3 3 3
2/1
3/2 4 2 1 2 3 5
Find maximum water flow with given capacities by eyes.

17. 3/3
1/1
1/1
1/1
5/3 3
3+3
2/1+1
3/2+1
4/3 2 1
2/2
3/2 5
Find maximum water flow with given capacities by eyes.

18. 3/3 5
1/1
1/1
1/1
5/3 3
3+3+1
2/1+1
3/2+1
4/3 2
1/1
2/2
3/2+1
5/1
Find maximum water flow 7 with given capacities by eyes.

19. 3/3
1/1
1/1
1/1 5
3/3 4 2 3
4/4
2/1 1
2/2
3/3
5/1
Flow 4 is maximal, but not a maximum flow.

20. 3/3
1/1
1/1
1/1 5
3/3 4 2 3
4/4
2/1 1
2/2
3/3
5/1
Blue path is an augmentative path with increasing value 2.

21. 3/3
1/1
1/1
1/1
5/2
3/3-2
4+2
2/2
3/2
4/4
2/1 1
2/2
3/3
5/1
Increasing or decreasing flow 2 in pipes according to augmentative path

22. 3/3
1/1
1/1
1/1
5/2
3/3-2
4+2+1
2/2
3/2+1
4/4
2/1-1
1/1
2/2
3/3
5/1
The same way to increase the pink flow 1, and to obtain the maximum flow 7.

23. Relation to optimization problemsc 3 b 1 1 1 5 3 2 3 a 4 d 2 1 2 3 5

24. Final comments
There always exists a trivial solution, 0 flow.
We don’t need to assume that xab is an integer, since each time we only increase integral flow value.
3. x<3 iff –x > -3
4. x<3 iff x+y=3 and y>0
5. Max x iff Min –x
6. Max xy is related to Max x+y, or Max x under the assumption x=cy.