Here are the details:
(1) We are given a number $f$ of units of flow that we wish to transmit from the source node to the sink node. For example, $f=5$ units of flow (data or goods) should be transmitted from the source node to the sink node.
(2) We are not given any plan or policy to transmit the flow. However, we know that $f$ is less than the maximum flow of the network. So, it is possible to transmit such an amount of flow through the network.
(3) We are given the arcs' capacities.
$\bullet$ Now, it is required to determine the residual network after sending these $f$ units of flow.
Input: A network with the given arcs' capacities and an amount of flow $f$. Also, two nodes are specified as source and sink nodes.
Output: A plan to transmit $f$ units of flow from the source node to the sink node, and the corresponding residual network.
It is clear that there may be more than one plan to send $f$ units of flow through the network, and so the residual network may change. It is not important to reach a special residual network. My main concern is only to find the time complexity of such a job (finding a way to transmit $f$ units of flow through the network, and then determining the residual network).
It is easy to see that if I have the plan to send $f$ units of flow, the residual network can be determined with time complexity of $O(m)$, where $m$ is the number of edges in the network.
$\bullet$ I highly appreciate any help in advance.