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Max Flow Min Cut

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Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity, then in a maximum flow every arc has integer flow.

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Find a maximum st-flow and a minimum st-cut s t

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Algorithm s t

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s t

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s t

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s t

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s t

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s t

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s t STOP

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Algorithm s t STOP

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Algorithm s t STOP

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Algorithm s t STOP

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Algorithm s t

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s t STOP

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Algorithm s t STOP

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Algorithm s t STOP

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Algorithm s t

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s t

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s t

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s t STOP

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Algorithm s t STOP

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Algorithm s t STOP

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Algorithm s t STOP

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Minimum Cut s t

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s t

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s t

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s t So f is a max flow

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Algorithm s t

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s t

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s t

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s t

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s t

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s t

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s t

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Hall’s Theorem from Max Flow Min Cut

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11111 Direct all edges from s to t and assign all arcs unit capacity Adds and t, adjacent to all of A and B respectively.

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We have to show that Hall’s Condition gives a 1-factor.

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A flow of value is enough to guarantee a 1-factor.

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So all we have to do is show for each cut S.

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