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Answer:
See explanation and answer below.
Step-by-step explanation:
The tranformation
For this case we need to construct G' dividing making a division for each vertex v of G into 3 edges that on this case are .
We assume that the edges from the begin are the incoming edges of and all the outgoing edges from v are outgoing edges from
We need to construct with capacity function a' and we need to satisfy the follwoing:
For every we create 2 vertices
Now we can add a new edge asscoiated to with the condition
Now for each edges we can create the following edge
and the capacity is given by:
And for this case we can see this:
Now we assume that x is the flow who belongs to G respect vertex capabilities. We can create a flow function x' who belongs to G' with the following steps:
For every edge we can assume that
Then for each vertex and we can define
and
And after see that the capacity constraint on this case would be satisfied since for every edge in G' on the form we have a corresponding edge in G because:
we have that:
And with this we have the maximization problem solved.
We assume that we have K vertices using the max scale algorithm.