1. Max Flow min Cut

2. Max Flow/Min Cut

3. Problem

4. Definitions
Capacity 7
Sink
Source

5. A Flow 3 5 7 2

6. A Cut

7. Capacity/Flow Across A Cut

8. Problem Find a cut with minimum capacity
Find maximum flow from source to sink

9. More Definition: Residual Graph3 5 7 2 5 2

10. Augmenting Path
A path from source to sink in the residual graph of a given flow

11. Idea
If there is an augmenting path in the residual graph, we can push more flow

12. Ford-Fulkerson Method
initialize total flow to 0
residual graph G’= G
while augmenting path exist in G’
pick a augmenting path P in G’
m = bottleneck capacity of P
add m to total flow
push flow of m along P
update G’

13. Example 3 2 1 3 1 3 1 1 3 1 1 4 3 2 2 1 1 1 2 4 2 4

14. Example 3 2 1 3 1 3 1 1 3 1 1 4 3 2 2 1 1 1 2 4 2 4

15. Example 3 2 1 3 1 3 1 1 3 1 1 4 3 2 2 1 1 1 2 3 1 3 1 1 1

16. Example 3 2 1 3 1 3 1 1 3 1 1 4 3 2 2 1 1 1 2 3 1 3 1 1 1

17. Example 1 2 1 3 1 1 1 1 1 1 1 4 3 2 2 1 1 1 2 3 1 3 1 1 1

18. Example 1 2 1 3 1 1 1 1 1 1 1 4 3 2 2 1 1 1 2 3 1 3 1 1 1

19. Example 1 2 1 3 1 1 1 1 1 1 1 3 3 2 1 1 1 1 2 2 2 2 2 2

20. Answer: Max Flow = 4 2 2 2 2 2 1 1 1 2 2 2

21. Answer: Minimum Cut = 4 3 2 1 3 1 3 1 1 3 1 1 4 3 2 2 1 1 1 2 4 2 4

22. Find Augmenting Path? initialize total flow to 0 residual graph G’= G
while augmenting path exist in G’
pick a augmenting path P in G’
m = bottleneck capacity of P
add m to total flow
push flow of m along P
update G’

23. Picking Augmenting Path
Edmonds-Karp’s Heuristics
Find a path with maximum bottleneck capacity
Find a path with minimum length

24. Variations
How about Maximum Cut?

25. Applications for MaxFlow/MinCut

26. Application: Bipartite Matching
Marriage Problem
BTW, how to determine if a graph is bipartite?

27. A Matching

28. Maximum Matching

29. Maximum Flow Problem

30. Maximum Flow/Matching

31. Minimum Vertex Cover Bipartite Graph5 2 1 2 4 3 2 3 4 2

32. A Vertex Cover (W=15) 5 2 1 2 4 3 2 3 4 2

33. Maximum Flow 5 2 1 2 4 3 2 3 4 2

34. Finite Cut = Vertex Cover5 2 1 2 4 3 2 3 4 2

35. Finite Cut = Vertex Cover

36. Problems..

37. Problem 1: Optimal Protein Sequence
Given a 2D geometric structure of a protein, with n residuals

38. Optimal Protein Sequence
Each residual can be either hydrophobic (H) or polar (P)

39. Optimal Sequence Each residual has a “solvent area”
Want to reduce solvent areas of Hs
Want to increase pairs of Hs in close contacts

40. Optimal Sequence Assign H, P to the residuals to minimize
d(i,j): 1 if H at position i and H at position j is close enough, 0 otherwise
s(i) : area exposed at position i

41. Problem 2: Forest Clearence
cannot clear
two adjacent
square!
profit for
cutting trees 5 10 8 12
which squares to clear to maximize profit?

42. Problem 3: All-Pair Maximum Bottleneck Bandwidth Path
Given a graph, how to find the maximum bottleneck bandwidth path between any two nodes, u and v, efficiently?