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2. www.monash.edu.au FIT2004 Algorithms & Data Structures L15: Flow Problems Prepared by: Bernd Meyer April 2007

3. www.monash.edu.au 3 Maximum Flow and Minimum Cut Max flow and min cut. –MaxFlow: determine the maximum possible flow from a source node a to a sink node b in some capacity-limited transport network –Min cut is dual to max flow. –Fundamental problems in combinatorial optimization. Many other problems reduce to max flow –scheduling (airlines, trains etc.) –transportation problems –matching problems –network routing –etc.

4. www.monash.edu.au 4 single commodity transport network. directed graph G = (V, E), no parallel edges. s = source, t = sink. c(e) = capacity of edge e. Flow Networks s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 capacity source sink

5. www.monash.edu.au 5 An s-t cut is a partition (A, B) of V with s  A and t  B. Its capacity is: Cuts Capacity = 10 + 5 + 15 = 30 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 A

6. www.monash.edu.au 6 An s-t cut is a partition (A, B) of V with s  A and t  B. Its capacity is: Cuts s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 A Capacity = 9 + 15 + 8 + 30 = 62

7. www.monash.edu.au 7 Find an s-t cut of minimum capacity. Minimum Cut Problem s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 A Capacity = 10 + 8 + 10 = 28

8. www.monash.edu.au 8 An s-t flow is a function that satisfies: –(capacity) For each e  E: –(conservation) For each v  V – {s, t}: The value of a flow f is: Flows Value = 4 4 0 0 0 0 0 0 4 4 0 0 0 0 capacity flow s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 4

9. www.monash.edu.au 9 An s-t flow is a function that satisfies: –(capacity) For each e  E: –(conservation) For each v  V – {s, t}: The value of a flow f is: Flows Value = 24 10 6 6 11 1 10 3 8 8 0 0 0 11 capacity flow s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 4

10. www.monash.edu.au 10 Find s-t flow of maximum value. Maximum Flow Problem 10 9 9 14 4 10 4 8 9 1 0 0 0 14 capacity flow s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 Value = 28

11. www.monash.edu.au 11 Flow value lemma. Let f be any flow, let (A, B) be any s-t cut. The net flow across the cut is equal to the amount leaving s. Flows and Cuts 10 6 6 11 1 10 3 8 8 0 0 0 11 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 Value = 24 4 A

12. www.monash.edu.au 12 Flow value lemma. Let f be any flow, let (A, B) be any s-t cut. The net flow across the cut is equal to the amount leaving s. Flows and Cuts 10 6 6 11 1 10 3 8 8 0 0 0 11 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 Value = 24 4 A

13. www.monash.edu.au 13 Flow value lemma. Let f be any flow, let (A, B) be any s-t cut. The net flow across the cut is equal to the amount leaving s. Flows and Cuts 10 6 6 11 1 10 3 8 8 0 0 0 11 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 Value = 24 4 A

14. www.monash.edu.au 14 Flows and Cuts Weak duality. Let f be any flow, and let (A, B) be any s-t cut. The value of the flow is at most the capacity of the cut. Cut capacity = 30  Flow value  30 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 Capacity = 30 A

15. www.monash.edu.au 15 Certificate of Optimality Corollary. Let f be any flow, let (A, B) be any s-t cut. If v(f) = cap(A, B), then f is a max flow and (A, B) is a min cut. Value of flow = 28 Cut capacity = 28  Flow value  28 10 9 9 14 4 10 4 8 9 1 0 0 0 14 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 15 4 4 0 A

16. www.monash.edu.au 16 Towards a Max Flow Algorithm Greedy algorithm. –Start with f(e) = 0 for all edge e  E. –Find an s-t path P where each edge has f(e) < c(e). –Augment flow along path P. –Repeat until you get stuck. s 1 2 t 10 00 00 0 20 30 Flow value = 0

17. www.monash.edu.au 17 Towards a Max Flow Algorithm Greedy algorithm. –Start with f(e) = 0 for all edge e  E. –Find an s-t path P where each edge has f(e) < c(e). –Augment flow along path P. –Repeat until you get stuck. s 1 2 t 10 200 0 30 Flow value = 20

18. www.monash.edu.au 18 Towards a Max Flow Algorithm greedy = 20 s 1 2 t 2010 20 30 200 0 opt = 30 s 1 2 t 2010 20 30 2010 20 This is only a local optimum!

19. www.monash.edu.au 19 Residual Graph Original edge: e = (u, v)  E. –Flow f(e), capacity c(e). Residual edge. –"Undo" flow sent. –e = (u, v) and e R = (v, u). –Residual capacity: Residual graph: G f = (V, E f ). –Residual edges with positive residual capacity. –E f = {e : f(e) 0}. uv 17 6 capacity uv 11 residual capacity 6 flow

20. www.monash.edu.au 20 Example 1 123 4 5 6 0/2 0/3 0/1 0/5 0/30/4 Augmenting path: 1→2 →3 →6 flow/capacity

21. www.monash.edu.au 21 123 4 5 6 2/2 0/3 0/1 2/5 0/30/4 Augmenting path: 1 →4 →3←2 →5 →6 Example 1 (cont.)

22. www.monash.edu.au 22 123 4 5 6 22 3 1 3 34 Augmenting path: 1 →4 →3←2 →5 →6 Example 1 (cont., Residual Graph) 2

23. www.monash.edu.au 23 123 4 5 6 2/2 1/3 1/1 1/5 1/31/4 max flow value = 3 Example 1 (maximum flow)

24. www.monash.edu.au 24 Augmenting Path Algorithm Augment(f, P) { b  bottleneck(P) foreach e  P { if (e  E) f(e)  f(e) + b else f(e R )  f(e) - b } return f } Ford-Fulkerson(G, s, t) { foreach e  E f(e)  0 G f  residual graph while (there exists augmenting path P) { f  Augment(f, P) update G f } return f } forward edge reverse edge

25. www.monash.edu.au 25 Augmenting Path Theorem Augmenting path theorem. Flow f is a max flow iff there are no augmenting paths. Max-flow min-cut theorem. [Ford-Fulkerson 1956] The value of the max flow is equal to the value of the min cut.

26. www.monash.edu.au 26 Ford-Fulkerson: Exponential Number of Augmentations Q. Is generic Ford-Fulkerson algorithm polynomial in input size? (m, n, and log C) A. No. If max capacity is C, then algorithm can take C iterations. s 1 2 t C C 00 00 0 C C 1 s 1 2 t C C 1 00 00 0 X 1 C C X X X 1 1 1 X X 1 1 X X X 1 0 1

27. www.monash.edu.au 27 Choosing Good Augmenting Paths Use care when selecting augmenting paths. –Some choices lead to exponential algorithms. –Clever choices lead to polynomial algorithms. Goal: –Can find augmenting paths efficiently. –Few iterations. Choose augmenting paths with: [Edmonds-Karp 1972, Dinitz 1970] –Max bottleneck capacity but can we find this cheaply? –Sufficiently large bottleneck capacity. –Shortest augmenting path (fewest edges)

28. www.monash.edu.au 28 Capacity Scaling Intuition. Choosing path with highest bottleneck capacity increases flow by max possible amount. –Don't worry about finding exact highest bottleneck path. –Maintain scaling parameter . –Let G f (  ) be the subgraph of the residual graph consisting of only arcs with capacity at least . 110 s 4 2 t 1 170 102 122 GfGf 110 s 4 2 t 170 102 122 G f (100)

29. www.monash.edu.au 29 Capacity Scaling Scaling-Max-Flow(G, s, t, c) { foreach e  E f(e)  0   smallest power of 2 greater than or equal to C G f  residual graph while (   1) { G f (  )   -residual graph while (there exists augmenting path P in G f (  )) { f  augment(f, P) update G f (  ) }    / 2 } return f }

30. www.monash.edu.au 30 Capacity Scaling: Running Time The scaling max-flow algorithm finds a max flow in O(m log C) augmentations. It can be implemented to run in O(m 2 log C) time.