Questions: Maximal Flow Problem. For the network shown below, determine the maximum flow from source node 1 to destination node 6. Flow capacities for each branch are indicated in the diagram. Once the three (3) network paths defined above are resolved, there are additional paths through the network by which flow can be increased between the input node and the output node.

Maximal Flow Problem. For the network shown below, determine the maximum flow from source node 1 to destination node 6. Flow capacities for each branch are indicated in the diagram. Once the three (3) network paths defined above are resolved, there are additional paths through the network by which flow can be increased between the input node and the output node.
Transcript text: Maximal Flow Problem. For the network shown below, determine the maximum flow from source node 1 to destination node 6 . Flow capacities for each branch are indicated in the diagram. Quiz: Assignment 6 - Networ canvas.vt.edu/courses/195038/quizzes/5... Question 4 1 pts Once the three (3) network paths defined above are resolved, there are $\qquad$ additional paths through the network by which flow can be increased between the input node and the output node. $\square$ - Previous Next ～
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Solution

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Solution Steps

Step 1: Identify the Source and Sink Nodes
  • The source node is 1.
  • The sink node is 6.
Step 2: List All Possible Paths from Source to Sink
  • Identify all paths from node 1 to node 6.
    • Path 1: 1 → 2 → 5 → 6
    • Path 2: 1 → 2 → 4 → 5 → 6
    • Path 3: 1 → 3 → 4 → 5 → 6
    • Path 4: 1 → 3 → 6
Step 3: Calculate the Flow for Each Path
  • Path 1: 1 → 2 → 5 → 6
    • Minimum capacity: min(17, 10, 4) = 4
  • Path 2: 1 → 2 → 4 → 5 → 6
    • Minimum capacity: min(17, 9, 10, 4) = 4
  • Path 3: 1 → 3 → 4 → 5 → 6
    • Minimum capacity: min(12, 14, 10, 4) = 4
  • Path 4: 1 → 3 → 6
    • Minimum capacity: min(12, 20) = 12

Final Answer

  • The maximum flow from node 1 to node 6 is the sum of the minimum capacities of the paths:
    • 4 (Path 1) + 4 (Path 2) + 4 (Path 3) + 12 (Path 4) = 24

The maximum flow through the network is 24.

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