To solve this problem using the network flow maximization algorithm, we'll follow these steps:

Initialize: Set the flow in all edges to $0.$

Find augmenting paths: Using a policy $($like BFS or DFS$),$ find an augmenting path from the source $($S$)$ to the sink $($t$)$ in the residual graph.

Augment flow: Along the augmenting path found, increase the flow by the maximum possible amount.

Update residual graph: Update the residual capacities of edges based on the flow added.

Repeat: Repeat steps $2-4$ until no augmenting path can be found.

Let's start with the initial setup of the flow graph $($Gf$)$ and the residual graph $($Gr$)$:

Initial Flow Graph $($Gf$)$:

$5$

S $---------->$ a

$/|\backslash |\backslash$

$0|\backslash 3|\backslash 2$

$|\backslash |\backslash$

$|\backslash |\backslash$

$|\backslash \_\_\_\_\_\_|\_\_\_\_\backslash$

$|6/6\backslash |6\backslash 5$

$|/\backslash ||$

$|/2||$

$|/||$

V$/$ V V

b $---------->$ d $---->$ t

$75$

Initial Residual Graph $($Gr$)$:

$5$

S $---------->$ a

$/|\backslash |\backslash$

$0|\backslash 3|\backslash 2$

$|\backslash |\backslash$

$|\backslash |\backslash$

$|\backslash \_\_\_\_\_\_|\_\_\_\_\backslash$

$|6/6\backslash |6\backslash 5$

$|/\backslash ||$

$|/2||$

$|/||$

V$/$ V V

b $---------->$ d $---->$ t

$75$

Now, let's find an augmenting path. We'll use BFS as the policy:

Start from S$.$

Visit neighbors with residual capacity $>0.$

Continue until reaching t$.$

One of the augmenting paths found using BFS is: S $->$ b $->$ d $->$ t$.$

Let's augment the flow along this path. The maximum flow we can add is limited by the minimum residual capacity along the path, which is $5.$

Updated Flow Graph $($Gf$)$:

$5$

S $---------->$ a

$/|\backslash |\backslash$

$0|\backslash 3|\backslash 2$

$|\backslash |\backslash$

$|\backslash |\backslash$

$|\backslash \_\_\_\_\_\_|\_\_\_\_\backslash$

$|1/6\backslash |6\backslash 5$

$|/\backslash ||$

$|/2||$

$|/||$

V$/$ V V

b $---------->$ d $---->$ t

$75$

Updated Residual Graph $($Gr$)$:

$0$

S $---------->$ a

$/|\backslash |\backslash$

$0|\backslash 3|\backslash 2$

$|\backslash |\backslash$

$|\backslash |\backslash$

$|\backslash \_\_\_\_\_\_|\_\_\_\_\backslash$

$|6/6\backslash |1\backslash 0$

$|/\backslash ||$

$|/2||$

$|/||$

V$/$ V V

b $---------->$ d $---->$ t

$70$

Now, we continue this process until no augmenting paths can be found or until reaching a desired maximum flow. The algorithm terminates when there are no augmenting paths from S to t in the residual graph.

This algorithm is guaranteed to terminate because with each iteration, the value of the flow increases until it reaches its maximum. It's also guaranteed to find the maximum flow because it keeps finding augmenting paths until no more paths can be found, and each augmenting path contributes to increasing the flow.

Explanation:

Let's delve deeper into each step of the algorithm and provide more explanation:

Initialization: At the start, we set the flow in all edges to $0.$ This sets up the initial state for the flow graph.

Finding Augmenting Paths: In this step, we use a policy, like Breadth$-$First Search $($BFS$)$ or Depth$-$First Search $($DFS$),$ to find a path from the source $($S$)$ to the sink $($t$)$ in the residual graph. The residual graph represents the remaining capacity in each edge after considering the current flow.

BFS Policy: BFS ensures that we find the shortest augmenting path in terms of the number of edges. This helps in efficiently finding augmenting paths and contributes to the algorithm's time complexity.

Augmenting Flow: Once we have found an augmenting path, we augment the flow along this path. The amount of flow we can add is limited by the minimum residual capacity along the path. This ensures that we don't exceed the capacity of any edge.

Updating Residual Graph: After augmenting the flow, we need to update the residual capacities of edges based on the flow added. This involves subtracting the flow from forward edges and adding the flow to backward edges to maintain the conservation of flow at each node.

Repeating the Process: Steps $2-4$ are repeated until no augmenting path can be found from S to t in the residual graph. This means that we have reached a maximum flow, and the algorithm terminates.

Termination Condition: The termination condition for the algorithm is when there are no augmenting paths from S to t in the residual graph. This indicates that we have reached a maximum flow, and further augmentations are not possible.

Correctness: The correctness of the solution lies in the fact that at each step, the algorithm either increases the flow along an augmenting path or terminates when no more augmenting paths can be found. Since each augmentation contributes to increasing the flow, and the algorithm terminates when no more augmenting paths exist, the maximum flow obtained