CSCC73H3 Lecture Notes - Flow Network, Minimum Cut, Stotting

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19 Oct 2011
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The label f /c on each edge indicates that the edge has capacity c, f of which is used by the ow. It is easy to check that the speci ed ow is proper (i. e. , it satis es the capacity and conservation constraints). V (f ) =pe out(s ) f (e) pe out(t ) f (e). By the max- ow-min-cut theorem, v (f ) =pe out(s ) c(e). Thus, by the capacity constraint, every edge in out(s ) is saturated in f . By assumption, v (f ) > 0, and so there is some e out(s ) such that f (e) > 0. As we just saw, such an edge is saturated in f , i. e. f (e) = c(e). Take f (e) = 0 for each edge e. this trivially satis es the capacity and conservation constraints. Thus it is an integral ow whose value is 0, as wanted.

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