1. Here is an example that involves what is called ‘back flow’ 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 Arrows have already been drawn initially showing the capacities and initial flows of zero along all of the edges. Network Flows – Back flow

2. To start with let’s choose the flow augmenting path SABT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 Network Flows – Back flow The minimum excess capacity along this path is 10, so we increase the flow by 10. Arrows in the direction of the path are decreased by 10 and arrows against the direction of the path are increased by 10. The minimum excess capacity along this path is 10, so we increase the flow by 10. Arrows in the direction of the path are decreased by 10 and arrows against the direction of the path are increased by 10.

3. To start with let’s choose the flow augmenting path SABT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 Network Flows – Back flow The arc AB is now saturated.

4. Our next flow augmenting path is SCDT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 Network Flows – Back flow The minimum excess capacity along this path is 15, so we increase the flow by 15. Arrows in the direction of the path are decreased by 15 and arrows against the direction of the path are increased by 15. The minimum excess capacity along this path is 15, so we increase the flow by 15. Arrows in the direction of the path are decreased by 15 and arrows against the direction of the path are increased by 15.

5. Our next flow augmenting path is SCDT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 Network Flows – Back flow Arc CD is now saturated.

6. Consider now the flow augmenting path SCBT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 Network Flows – Back flow The minimum excess capacity along this path is 8, so we increase the flow by 8. Arrows in the direction of the path are decreased by 8 and arrows against the direction of the path are increased by 8. The minimum excess capacity along this path is 8, so we increase the flow by 8. Arrows in the direction of the path are decreased by 8 and arrows against the direction of the path are increased by 8.

7. Consider now the flow augmenting path SCBT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 Network Flows – Back flow Arc SC is now saturated.

8. Another flow augmenting path is SADT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 7 13 3 0 18 6 Network Flows – Back flow The minimum excess capacity along this path is 3, so we increase the flow by 3. Arrows in the direction of the path are decreased by 3 and arrows against the direction of the path are increased by 3. The minimum excess capacity along this path is 3, so we increase the flow by 3. Arrows in the direction of the path are decreased by 3 and arrows against the direction of the path are increased by 3.

9. Another flow augmenting path is SADT. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 7 13 3 0 18 6 Network Flows – Back flow Arc DT is now saturated.

10. Are there any more flow augmenting paths? 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 7 13 3 0 18 6 Remember that a flow augmenting path is any path from S to T following non-zero arrows. The path SADCBT is such a path. Network Flows – Back flow Although arc CD is saturated, in this path we are going against the direction of the arc – hence the term back flow.

11. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 7 13 3 0 18 6 What is actually happening is that flow along CD is being diverted along CB which means that DT is able to take more flow from AD as a result. 5 15 4 5 13 2 0 10 1 20 Network Flows – Back flow The minimum excess capacity along SADCBT is 2 and so the flow changes by 2. All arrows in the direction of the path are reduced by 2, All arrows against the direction of the path are increased by 2. The minimum excess capacity along SADCBT is 2 and so the flow changes by 2. All arrows in the direction of the path are reduced by 2, All arrows against the direction of the path are increased by 2.

12. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 7 13 3 0 18 6 What is actually happening is that flow along CD is being diverted along CB which means that DT is able to take more flow from AD as a result. 5 15 4 5 13 2 0 10 1 20 Network Flows – Back flow Arc CB is now saturated, and arc DC is no longer saturated.

13. 0 A C B D 20 10 23 21 18 15 10 9 S T 0 0 0 0 0 0 0 10 10 10 11 10 0 8 15 15 3 150 0 23 8 3 18 2 3 17 3 0 18 6 We have a flow of 23 + 15 = 38 out of the source and 20 + 18 = 38 into the sink. This is the maximum flow. The flows are equal to all the arrows going against the direction of the edges. We have a flow of 23 + 15 = 38 out of the source and 20 + 18 = 38 into the sink. This is the maximum flow. The flows are equal to all the arrows going against the direction of the edges. 5 15 4 5 13 2 0 10 1 20 10 13 23 20 15 5 10 18 Network Flows – Back flow