mthsc810-lecture19-2x2

mthsc810-lecture19-2x2 - MthSc 810 Mathematical Programming...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 19 Pietro Belotti Dept. of Mathematical Sciences Clemson University December 1, 2011 Reading for today: Sections 7.5 Reading for Tuesday: Sections 6.1, 6.2. Cuts ◮ An s − t cut of a (flow) network G = ( V , A ) 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 ) ◮ The cut’s capacity is c ( S , T ) = ∑ u ∈ S ∑ v ∈ T c ( u , v ) S T ( u 1 , v 1 ) ( u 2 , v 2 ) ( u 3 , v 3 ) ( u k , v k ) s t a b 2 / 2 6 / 4 3 5 / 2 4 / 4 e.g. f ( S , T ) = 2 + 4 = 6; c ( S , T ) = 2 + 3 + 4 = 9 ◮ A minimum cut of G is a cut whose capacity is minimum among all cuts A Simple Upper Bound Lemma For any cut ( S , T ) , f ( S , T ) = | f | Proof : f ( S , T ) = f ( S , V ) − f ( S , S ) Since S ∪ T = V , S ∩ T = ∅ = f ( S , V ) = f ( { s } , V ) + f ( S \ { s } , V ) flow conservation = f ( { s } , V ) = | f | Corollary The value of a flow is no more than the capacity of any cut | f | = f ( S , T ) = X u ∈ S X v ∈ T f ( u , v ) ≤ X u ∈ S X v ∈ T c ( u , v ) = c ( S , T ) . Residual Network ◮ Given a flow f in a network G = ( V , A ) , how much more flow can be pushed from u ∈ V to v ∈ V ? ◮ Easy: the residual capacity of the arc ( u , v ) , c f ( u , v ) def = c ( u , v ) − f ( u , v ) ≥ . ◮ Given flow f , create a residual network G f = ( V , A f ) , with A f def = { ( u , v ) ∈ V × V | c f ( u , v ) > } ◮ Each arc in G f can admit a positive flow. s t a b 2 / 2 6 / 4 3 5 / 2 4 / 4 s t a b 2 4 2 3 3 2 4 Augmenting Flow Lemma ◮ We define the flow sum of two flows f 1, f 2 as the sum of the individual flows ( f 1 + f 2 )( u , v ) = f 1 ( u , v ) + f 2 ( u , v ) . ◮ Note that f 1 + f 2 is also a flow function Augmenting Flow Lemma Given a flow network G , a flow f in G . Let f ′ be any flow in the residual network G f . Then the flow sum f + f ′ is a flow in G with value | f | + | f ′ | Augmenting Paths ◮ Consider a path P st from s to t in G f ....
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This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

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mthsc810-lecture19-2x2 - MthSc 810 Mathematical Programming...

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