This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.
 Fall '08
 Staff
 Math

Click to edit the document details