Network Analysis

1. Network Analysis Maxflow

2. What is a Network? • Directed connected graph • Source node • Sink (destination node) • Arcs are weighted (costs) • Represent a system

3. Maxflow problem • Determine the greatest flow through the network from Source to Sink. • Source generates flow • Sink consumes flow • Intermediate nodes only pass flow • Maximum capacity on each link

4. Example: Trams • Network represents tram routes • Weights are number of free places • How many people can get from A to G?

5. Maxflow: notations • n = the number of nodes • node 1 = Source • Node n = Sink • xij = the ammount of flow on arc from i to j • uij = capacity on arc from i to j

6. Linear Programming

7. Algorithm • Simplex method • Take advantage of special network structure • Ford-Fulkerson labeling algorithm • Based on flow-augmenting-paths = path from source to sink whose current flow is less than its capacity.

8. Ford-Fulkerson labeling algorithm • Use a labeling procedure to find a flow-augmenting path. If none can be found, the current flow is optimal • Increase the current flow as much as possible in the flow-augmenting path (until some arc reaches its capacity) Goto 1.

9. Labeling Procedure (1) • Label the Sourcenode • For any labeled node i examine: • outgoing arcs (i,j) • incoming arcs (j,i)

10. Labeling Procedure (2) • Outgoing arc : label node j if the current flow is less than its capacity (uij) • Incoming arc : label node j if the incoming flow is greater than zero

11. Example (1) • Assume flow in all xij= 0 • Label nodes A,B,C,D and G • Find flow-augmenting path (A,D)(D,G) • Increase flow by 4 • f = 4

12. Example (2) • Label nodes A,B,C and E,D,F,G • Find flow-augmenting path (A,B)(B,D)(D,E)(E,G) • Increase flow by 4 • f = 8

13. Example (3) • Label nodes A,B,C and E,D,F,G • Find flow-augmenting path (A,D)(D,F)(F,G) • Increase flow by 2 • f = 10

14. Example (4) • Label nodes A,B,C,D,F and G • Find flow-augmenting path A,C,D,F,G • Increase flow by 4 • f = 14

15. Example (5) • Label all nodes except G • No more flow-augmenting paths! • Current flow f= 14 is optimal. • Question: Why is D labeled?

16. Backward Arcs (1) • Assume path 1,2,4,6 • f = 4 • The use of backward arcs is necesarry to find the max-flow! • Why?

17. Backward Arcs (2) • f = 8 due to backward labeling! • Flow trough backward arcs is subtracted from total flow.

18. Max flow – Min Cut (1) • A cut is a set of arcs that, when removed, make the Sink unreachable from Source. • Whithout a cut, there is no flow from Source to Sink. • Capacities of a cut limit the Max flow • A min-cut is a cut with minimum total capacity.

19. Max flow – Min Cut (2) • Ford and Fulkerson’s “Max flow-Min Cut” theorem • Max flow = Min Cut • If no flow-augmenting path can be found, it is proven that the maximum capacity of some cut is used.

20. Extensions to Max flow • Multiple Sources : include supersource • Multiple Sinks : include supersink • Limited Source(s) • Limited Sink(s)

21. Multiple Sources • Supersource connects to all sources with unrestricted capacity. • Uncapacitated links do not contibute to Min Cut!

22. Multiple Sinks • All Sinks connect to Supersink with unrestricted capacity • Uncapacitated links do not contribute to Min Cut!

23. Limited Sources/Sinks • Include supersource/sink • Place capacity on connections between supersource(s)/sink(s) and source(s)/sink(s)

24. Complexity of Max flow • Depends on how the flow-augmenting paths is selected. • Shortest path method: O(ne²) • Blocking flows method: O(n²e) • Preflow method: O(n³)