Prove or disprove: If \((A, V-A)\) and \((B, V-B)\) are both minimum capacity st-cuts in a network, is \((A \cap B, V - (A \cap B))\) also a minimum capacity st-cut?
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In a network, an st-cut is a partition of the vertices into two sets, A and B, such that the source vertex s is in A and the sink vertex t is in B. The capacity of an st-cut is the sum of the capacities of the edges crossing the cut from A to B. Now, let's Show more…
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Recall that a flow network is a directed graph G = (V, E) with a source s, a sink t, and a capacity function c : V × V → ℝ⁺₀ that is positive on E and 0 outside E. We only consider finite graphs here. Also, note that every flow network has a maximum flow. This sounds obvious but requires a proof (and we did not prove it in the video lecture). Which of the following statements are true for all flow networks (G, s, t, c)? If G = (V, E) has as cycle then it has at least two different maximum flows. (Recall: two flows f, f' are different if they are different as functions V × V → R. That is, if f(u, v) ≠ f'(u, v) for some u, v ∈ V. The number of maximum flows is at most the number of minimum cuts. The number of maximum flows is at least the number of minimum cuts. If the value of f is 0 then f(u, v) = 0 for all u, v. The number of maximum flows is 1 or infinity. The number of minimum cuts is finite.
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