Task $1$

Given the following network:

Here the numbers on the edges denote the number of parallel connected cables for each connection where single cable has capacity of $1Gbps.$ The server $Q$ transfers data to the server $E$ via given routing table.

a$)$ Assume that there is no bottleck between routers $(2-5),$ what is the potential maximum data rate that servers $Q$ and $E$ would support?

b$)$ Determine the actual maximum data rate that Server $Q$ can achieve when sending data to Server E over the routing network. Use the Ford$-$Fulkerson algorithm for this.

c$)$ Determine the associated residual graph and remove the irrelevant edges.

d$)$ Do the data cables in the routing network form a bottleneck? Based on your results, determine the minimum cut of the routing network by specifying the complementary sets $S$ and $S^c$ as well as the edges involved in the minimum cut. Then compare the result with the result from a$).$

e$)$ Now, you are allowed to disconnect some of the cables and reuse them to achieve a higher rate. However, you are not allowed to disconnect the cables connected to the servers, i$.$e$.,$ the cables between, $(Q2),(Q4),(3E),$ and, $(5E).$ Which cables will you disconnect. Draw the new outlook of the network and calculate the achievable data rate.