Lecture12 - Max-Flow / Min-Cut Theorem MS&E...

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Max-Flow / Min-Cut Theorem May 22, 2006
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Refresher on the Duality Theorems r The Weak Duality Theorem tells us that any solution to the dual gives us an upper (lower) bound r The Strong Duality Theorem tells us that at the optimal solution (if it exists) for both the primal and the dual, the objective values are exactly the same
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Duality and Maximum Flow Problems r Remember: Maximum Flow Problems are Linear Programs—they just so happen to be special LPs because under the right settings, the solutions are INTEGRAL c They are special cases of the Min-cost problem r So, what will the duality theorems tell us about Max Flow Problems?
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Max Flow Formulation s t 2 10 5 4 2 3 7 r Example from last lecture
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r Example from last lecture r Add an Auxiliary edge (dummy edge) with infinite capacity from t to s. Maximize flow on this edge Max Flow Formulation s t 2 10 5 4 2 3 7
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Max Flow Formulation ts
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Example s d f e t 2 10 5 4 2 3 7 g a b c 3 1 4 5 3 1 1 8
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Example (with Auxiliary Edge) s d f e t 2 10 5 4 2 3 7 g a b c 3 1 4 5 3 1 1 8
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Example—Excel Solution s d f e t 2/2 9/10 5/5 0/4 2/2 3/3 3/7 g a b c 1/3 1/1 4/4 4/5 2/3 1/1 1/1 3/8 11/∞ Total Flow=11
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This note was uploaded on 09/23/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

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Lecture12 - Max-Flow / Min-Cut Theorem MS&E...

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