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Max flow

How many arcs are in set Q (all non existing arcs)?

To determine the number of arcs in set \( Q \), which represents all non-existing arcs in the graph, we need to analyze the directed graph given and compare it to a complete graph.

### Steps to Find Set \( Q \)

1. **Count All Possible Arcs**: For a directed graph with \( n \) nodes, a complete graph has \( n(n - 1) \) directed arcs because each node can have an arc to every other node except itself.
   
   In this case, there are 8 nodes (A, B, C, D, E, F, G, H), so the total possible arcs are:
   \[
   8 \times (8 - 1) = 56
   \]

2. **Count Existing Arcs**: We count all arcs present in the graph.
   - From the diagram:
     - A → B, A → C
     - B → C, B → E
     - C → D
     - D → F, D → H
     - E → F, E → G
     - F → H
   - Total existing arcs: 11

3. **Calculate Non-Existing Arcs in Set \( Q \)**:
   \[
   |Q| = 56 - 11 = 45
   \]

So, the number of arcs in set \( Q \) is **45**.

Would you like a further breakdown of any step, or have any additional questions? Here are some related questions for further exploration:

1. How would you model the capacity constraints for each arc in the LP formulation?
2. What objective function would you set to maximize flow from node A to node H?
3. How would you write the flow conservation constraints for intermediate nodes like B, C, D, etc.?
4. Could you discuss the effect of removing an arc on the maximum flow?
5. How would the LP change if you wanted to minimize the flow instead?

**Tip:** When analyzing flow problems, always verify if arcs have capacity constraints, as these directly impact the formulation and solution of the max flow problem.

Write an LP for this max flow problem from Node A to H

Solve a max flow problem using linear programming to maximize flow from Node A to Node H in a directed network. Explore concepts like flow conservation, capacity constraints, and the max-flow min-cut theorem.

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