1. Flow Networks zichun@comp.nus.edu.sg

2. Formalization Basic Results Ford-Fulkerson Edmunds-Karp Bipartite Matching Min-cut

3. Flow Network

4. Properties of Flow

5. Maximum Flow Value of Flow

6. Motivating Problem s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 16 10 4 12 20 9 7 4 14 13

7. Motivating Problem s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 15/20 4/9 7/7 4/4 11/14 8/13

8. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 15/20 4/9 7/7 4/4 11/14 8/13 12/12

9. Multiple Sources / Sinks s1s1 v1v1 v2v2 v3v3 v4v4 t1t1 16 10 4 12 20 9 7 4 14 13 t2t2 4 s2s2 4

10. Multiple Sources / Sinks s1s1 v1v1 v2v2 v3v3 v4v4 t1t1 16 10 4 12 20 9 7 4 14 13 t2t2 4 s2s2 4 S t

11. Implicit Summation Notation

12. Key Equalities

14. Flow Value Definition Homomorphism Flow Conservation Skew Symmetry Homomorphism Flow Conservation

15. Ford-Fulkerson

16. Residual network

17. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 15/20 4/9 7/7 4/4 11/14 8/13 s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5

18. Augmenting Path A path of non-zero weight from s to t in G f

19. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 15/20 4/9 7/7 4/4 11/14 8/13 s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5

20. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 15/20 4/9 7/7 4/4 11/14 8/13 s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5

21. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 15/20 4/9 7/7 4/4 11/14 8/13 s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5

22. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 20/20 0/9 7/7 4/4 11/14 13/13 s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 20 4 11 13 11 0 7 0 3 0 9 3 0 0 0

23. S-T Cut

24. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11/16 10 1/4 12/12 15/20 4/9 7/7 4/4 11/14 8/13 f(S, T) = 12 – 4 + 11 = 19 c(S, T) = 12 + 14 = 26

25. Let f be a flow in a flow network G with source s and sink t, and let (S, T) be a cut of G. Then the net flow across (S, T) is f(S, T) = |f| Homomorphism Flow Conservation Homomorphism Flow Conservation Definition

26. The value of any flow f in a flow network G is bounded by the capacity of any cut of G

27. Min-Cut Max-Flow 1. f is a maximum flow in G 2.The residual network G f contains no augmenting path 3.|f|= c(S, T) for some cut (S, T) of G

28. 1.Premise: f is a max-flow in G 2.Assume G f has augmenting path p 3.We can augment G f with p to get a flow f’ > f – Contradicts [1] Hence G f has no augmenting paths

30. The value of any flow f in a flow network G is bounded by the capacity of any cut of G

31. Ford-Fulkerson Termination: G f has no augmenting path iff flow is maximum

32. Run-time

33. s v1v1 v2v2 t 1 1,000,000

34. s v1v1 v2v2 t 1

35. s v1v1 v2v2 t 1 999,9991,000,000 1 1

36. s v1v1 v2v2 t 1 999,9991,000,000 1 1

37. s v1v1 v2v2 t 1 999,999 1,000,000 999,999 1 1 1 1

38. s v1v1 v2v2 t 1 1,000,000 999,999 1 1 1 1

39. Edmunds Karp Run-time: O(VE 2 )

40. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5 Critical Edge

41. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5 Critical Edge

42. s v1v1 v2v2 v3v3 v4v4 t Factory Warehouse 11 5 12 15 4 11 8 4 7 5 3 0 5 3 0 0 5

44. Since there are O(E) edges, the number of augmenting path is bounded by O(VE) [by Lemma]. Run-time: O(VE 2 )

45. Bipartite Matching

47. st

48. st