Updating maximum flow

Maximum Flow: It is defined as the maximum amount of flow that the network would allow to flow from source to sink. A demonstration of working of Ford-Fulkerson algorithm is shown below with the help of diagrams.

Multiple algorithms exist in solving the maximum flow problem. Implementation: The complexity of Ford-Fulkerson algorithm cannot be accurately computed as it all depends on the path from source to sink.

updating maximum flow-38

Updating maximum flow

A residual network graph indicates how much more flow is allowed in each edge in the network graph.

function: Ford Fulkerson(Graph G, Node S, Node T): Initialise flow in all edges to 0 while (there exists an augmenting path(P) between S and T in residual network graph): Augment flow between S to T along the path P Update residual network graph return An augmenting path is a simple path from source to sink which do not include any cycles and that pass only through positive weighted edges.

A pseudocode for this algorithm is given below, Inputs required are network graph $$G$$, source node $$S$$ and sink node $$T$$.

Instead, if path chosen are only $$S-A-T$$ and $$S-B-T$$, would also generate the maximum flow. It includes construction of level graphs and residual graphs and finding of augmenting paths along with blocking flow.

Level graph is one where value of each node is its shortest distance from source.

Blocking flow includes finding the new path from the bottleneck node.

Residual graph and augmenting paths are previously discussed. Inputs required are network graph G, source node S and sink node T.

function: Dinic Max Flow(Graph G, Node S, Node T): Initialize flow in all edges to 0, F = 0 Construct level graph while (there exists an augmenting path in level graph): find blocking flow f in level graph F = F + f Update level graph return F Update of level graph includes removal of edges with full capacity.

Removal of nodes that are not sink and are dead ends.

A demonstration of working of Dinic's algorithm is shown below with the help of diagrams.

I'm given a Control Flow Graph with natural numbers representing edge capacites and a max flow found by running Ford Fulkerson. My question is this: Why is this different than just running FF with the original graph, but changing the capacity of e to e+1?

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