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I am wondering if the celebrated duality between max-flow and min-cut actually tolerates infinite valued capacities. Here is a simple example where it seems not:

source s, sink t, five other nodes a, b, c, d, e

s -> a: capacity 3

s -> b: 3

a -> c: \infty

a -> d: \infty

b -> d: \infty

b -> e: \infty

c -> t: 1

d -> t: 1

e -> t: 4

The max flow is 5. However, there is no cut whose capacity is 5. This is because the infinite capacities force all a, b, c, d, e to belong to the same set/half of a cut (otherwise there would be an \infty weight in the cut-set).

Janathan
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2 Answers2

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oh, I forgot that when the graph is directed, for an edge (u, v) to be counted into the cut weight, not only should u and v belong to different halves of the cut, but also u should be in the same half as the source s, and v in the same half as the sink t.

So now there is a trivial cut with capacity 5: S = {s, a, c, d} T = {b, e, t}

Janathan
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It does, but only when there is at least one cut with finite capacity. Otherwise, as your example shows, it gives no information about the maximum flow.

Rindy
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