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lecture25 - Taking Stock IE170 Algorithms in Systems...

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IE170: Algorithms in Systems Engineering: Lecture 25 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University March 30, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 25 Lecture Notes 1 / 23 Taking Stock Last Time Flows This Time (Cardinality) Matching Homework and Review Jeff Linderoth (Lehigh University) IE170:Lecture 25 Lecture Notes 2 / 23 Flows The Big Kahuna Max-Flow Min-Cut Theorem The following statements are equivalent 1 f is a maximum flow 2 f admits no augmenting path. (No ( s, t ) path in residual network) 3 | f | = c ( S, T ) for some cut ( S, T ) Jeff Linderoth (Lehigh University) IE170:Lecture 25 Lecture Notes 3 / 23 Flows Ford-Fulkerson Algorithm This gave Lester Ford and Del Fulkerson an idea to find he maximum flow in a network: Ford-Fulkerson ( V, E, c, s, t ) 1 for i 1 to n 2 do f [ u, v ] f [ v, u ] 0 3 while augmenting path P in G f 4 do augment f by c f ( P ) Assume all capacities are integers. If they are rational numbers, scale them to be integers. Jeff Linderoth (Lehigh University) IE170:Lecture 25 Lecture Notes 4 / 23
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Flows Analysis If the maximum flow is | f | * , then (since the augmenting path must raise the flow by at least 1 on each iteration), we will require ≤ | f | * iterations. Augmenting the flow takes O ( | E | ) Ford-Fulkerson runs in O ( | f | * | E | ) This is not polynomial in the size of the input.
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