Please do this solution properly and give me the solution
Useless Nodes
It is reasonable to restrict our flow networks to graphs G = (V, E) where every node u is on some path from the source to the sink. That is, for all u ∈ V, s → u → t. Suppose that we have a flow network where this is not the case. That is, there are some useless nodes that are not on any path from the source s to the sink t. Intuitively, useless nodes could not be part of any maximum flow because any flow sent from s to u would get stuck, thus violating flow conservation somewhere. To support this intuition, we want to show that for any flow network G, there is a maximum flow in G where the flow into and the flow out of every useless node is zero. Let G = (V, E) be any flow network. We define the following sets: A = {u ∈ V | s and u → t} // u is on a path from s to t X = {u ∈ V | s but u → t} // u is reachable from s, but cannot reach t Y = {u ∈ V | s and u → t} // u is not reachable from s, and cannot reach t Z = {u ∈ V | u → t but s ∼ u} // u can reach t, but is not reachable from s
Note that A, X, Y, and Z are pairwise disjoint. The nodes in X, Y, and Z are all useless by our definition. Also, s and t are nodes in A.
Assignment:
1. Give an example of a flow f in a flow network G where the flow into and out of some nodes in Y are non-zero. Make sure your flow f satisfies flow conservation.
2. Argue that despite your example above, for every flow network G, there is a maximum flow where the flow into and the flow out of every u ∈ Y is zero.
3. Recall that for a cut (S, T), the net flow across this cut is denoted by f(S, T):
CS - CT
CS - CT
Furthermore, recall that |f| is the flow value and that we showed in the proof of the Max Flow Min Cut Theorem that for any cut (S, T).
f = f(S, T)
Consider the cut (S, T) where S = A ∪ Y ∪ Z ∪ {t} and T = X ∪ {t}. Use the formulas above to show that the flow into X must be zero. I.e.,
∀u ∈ S, ∀v ∈ X, f(u, v) = 0.
Hint: first show that Σf(a, v) = 0. u ∈ S, v ∈ X
Also, for convenience, assume that there are no edges into the source s and no edges out of the sink t.
4. Argue that for any flow network G, there is a maximum flow where the flow into and the flow out of every u ∈ X is zero.
5. Argue that for any flow network G, there is a maximum flow where the flow into and the flow out of every useless node is zero. (Thus, conclude that, as far as the maximum flow is concerned, useless nodes are indeed useless.)
