lecture24 - IE170: Algorithms in Systems Engineering:...

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Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 24 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University March 28, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 1 / 24 Taking Stock Last Time Transitive Closure (Fast) Flows in Networks This Time Flows, Flows, Flows Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 2 / 24 Flows Flows in Networks G = ( V,E ) directed. Each edge ( u,v ) E has a capacity c ( u,v ) If ( u,b ) E c ( u,v ) = 0 We will typically have a special source vertex s V , a sink vertex t V , and we will assume there exists paths from s v t v V The combination of all of these things ( G,s,t,c ) is known as a flow network . Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 3 / 24 Flows Net Flows A net flow is a function f : V V R | V || V | that satisfies three conditions: 1 Capacity Constraints: f ( u,v ) c ( u,v ) 2 Skew Symmetry: f ( u,v ) =- f ( v,u ) u V,v V 3 Flow Conservation: v V f ( u,v ) = 0 u V \ { s,t } Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 4 / 24 Flows More Flow An important value we will be worried about is the value of flow f = | f | = v V f ( s,v ) : The total flow out of the source. The Maximum Flow Problem Given G = ( V,E ) . source node s V , sink node t V , edge capacities c . Find a flow whose value is maximum. Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 5 / 24 Flows Lemma, Lemma, Lemma Recall Shorthand f ( X,Y ) = x X y Y f ( x,y ) . 1 f ( X,X ) = 0 X V 2 f ( X,Y ) =- f ( Y,X ) X,Y V 3 Let X,Y,Z V be such that X Y = , then f ( X Y,Z ) = f ( X,Z ) + f ( Y,Z ) f ( Z,X Y ) = f ( Z,X ) + f ( Z,Y ) 4 | f | = f ( V,t ) Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 6 / 24 Flows Cuts A cut of a (flow) network G = ( V,E ) is a partition of V into S and T = V \ S such that s S and t T For flow f , net flow across a cut is f ( S,T ) and the cuts capacity is c ( S,T ) = u S v T c ( u,v ) A minimum cut of G is a cut whose capacity is minimum Jeff Linderoth (Lehigh University) IE170:Lecture 24 Lecture Notes 7 / 24 Flows A Simple Upper Bound...
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This note was uploaded on 08/06/2008 for the course IE 170 taught by Professor Ralphs during the Spring '07 term at Lehigh University .

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lecture24 - IE170: Algorithms in Systems Engineering:...

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