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Question 8.1. Find a maximum st-flow in the network shown in Figure 8.3, starting with the given flow f consisting of unit flow in the st-path of length four at the top of the diagram. Also find a minimum cut in the network. The capacities of the arcs are denoted by numbers next to each arc. Figure 8.3: Network As we stated, the proof of the Max-Flow Min-Cut Theorem gives an algorithm for finding a maximum flow as well as a minimum cut. To construct a maximum flow f and a minimum cut (S,S), proceed as follows: start by letting f be the zero flow and S={s} where s is the source. Construct a set S as in the theorem: whenever there is an arc (x,y) such that f(x,y)0 and xS and y/S, add y to S. If at the end of this procedure, t/S, then let S=S to get a minimum cut and the current flow is a maximum flow. If at the end of this procedure tS, then there must be a path x0x1x2,xr where s=x0 and t=xr, along which f can be augmented by some value >0. The value of is given in the proof above: it is =min{c(xi,xi+1)f(xi,xi+1),f(xi+1,xi)0if(xi,xi+1) and f(xi+1,xi) if f(xi+1,xi)>0, for each i:0i