mthsc810-lecture19-2x2

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

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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 ) = u S v T f ( u , v ) u S 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 ) 0 . Given flow f , create a residual network G f = ( V , A f ) , with A f def = { ( u , v ) V × V | c f ( u , v ) > 0 } 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
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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 . According to the lemma, we can increase the flow in G by increasing the flow along arcs in P st a sequence of pipes s t with an extra capacity How much more? c f ( P st ) = min { c f ( u , v ) | ( u , v ) is on path P st } . Augmenting Paths Augmenting flow: Let P be an augmenting path in G f , define f P : V × V R : f P ( u , v ) = c f ( P ) ( u , v ) on P c f ( P ) ( v , u ) on P 0 otherwise then f P is a flow in G f with value | f P | = c f ( P ) > 0 Corollary: f = f + f P is a flow in G with value | f
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