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Lecture12 - Max-Flow Min-Cut Theorem MS&E 111-Introduction...

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Max-Flow / Min-Cut Theorem MS&E 111—Introduction to Optimization May 22, 2006
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Refresher on the Duality Theorems ring2 The Weak Duality Theorem tells us that any solution to the dual gives us an upper (lower) bound ring2 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 ring2 Remember: Maximum Flow Problems are Linear Programs—they just so happen to be special LPs because under the right settings, the solutions are INTEGRAL circle6 They are special cases of the Min-cost problem ring2 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 ring2 Example from last lecture
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ring2 Example from last lecture ring2 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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