Exploring Network Flow Problems with the Ford-Fulkerson Method

What is the Ford-Fulkerson Method?

The Ford-Fulkerson method is a popular algorithm used to compute the maximum flow in a flow network. It operates by finding augmenting paths in the residual graph and increasing the flow until no more augmenting paths can be found.

Key Concepts

• Flow Network: A directed graph where each edge has a capacity and each edge receives a flow.
• Source (s): The starting point of flow in the network.
• Sink (t): The endpoint where the flow is directed.
• Capacity: The maximum amount of flow that an edge can support.
• Residual Capacity: The remaining capacity of an edge after accounting for the flow sent.

Example Problem

Consider the following flow network:

    (s)
     / 
    10  
   /   
  (A)----->(B)
   |  5   / | 
   |      /  | 15 
   |     /   |    

    ----->(t) 

• Vertices: s (source), A, B, t (sink)
• Edges and Capacities:
  • (s, A): 10
  • (s, B): 5
  • (A, B): 15
  • (A, t): 10
  • (B, t): 10

Step 1: Initialize Flows

Initially, all flows are set to 0.

Step 2: Find Augmenting Paths

We can find a path from source s to sink t:

• Path 1: s -> A -> t (minimum capacity is 10)

Flow Update:

• The flow along this path is increased by 10. The new flow values are:
  • (s, A): 10
  • (A, t): 10

Step 3: Update Residual Capacities

The residual capacities are updated:

• (s, A): 0 (10 - 10)
• (A, t): 0 (10 - 10)
• (s, B): 5
• (A, B): 15
• (B, t): 10

Step 4: Find Another Augmenting Path

Now we look for another path:

• Path 2: s -> B -> t (minimum capacity is 5)

Flow Update:

• The flow along this path is increased by 5:
  • (s, B): 5
  • (B, t): 5

Step 5: Update Residual Capacities Again

The residual capacities are now:

• (s, A): 0
• (A, t): 0
• (s, B): 0 (5 - 5)
• (A, B): 15
• (B, t): 5 (10 - 5)

Summary of Flows

At this point, the flows are:

• (s, A): 10
• (s, B): 5
• (A, t): 10
• (B, t): 5

Maximum Flow Calculation

The maximum flow from source s to sink t in this network is:

• Total flow = Flow from A to t + Flow from B to t = 10 + 5 = 15.

Conclusion

The Ford-Fulkerson method effectively helps in determining the maximum flow in a flow network. By following systematic steps and updating flows and capacities, we can solve complex network flow problems with clarity. Understanding the basics of capacity, flow, and residual graphs is essential for applying this method successfully.