Lecture12

# 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
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
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
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
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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