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Unformatted text preview: 3.2 Reductions Among Different Flow Models 103 mean the obvious thing when b ( v ) = 0 and if b ( v ) > 0 ( b ( v ) < 0) we think of l * ( v ) ,u * ( v ) ,c * ( v ) as bounds and costs per unit on the total amount of flow out of (in to) v . Let D ST be the digraph obtained from D = ( V,A ) by performing the vertex splitting procedure. Define a new network based on the digraph D ST by adding lower bounds, capacities and costs as follows: (a) For every arc i s j t (corresponding to an arc ij of A ) we let h ( i s j t ) = h ( ij ), where h ∈ { l,u,c } . (b) For every arc i t i s (corresponding to a vertex i of V ) we let h ( i t i s ) = h * ( i ), where h * ∈ { l * ,u * ,c * } . Finally we define the function b as follows: If b ( i ) = 0 then b ( i s ) = b ( i t ) = 0; If b ( i ) > 0 then b ( i t ) = b ( i ) and b ( i s ) = 0; If b ( i ) < 0 then b ( i t ) = 0 and b ( i s ) = b ( i ) . 1 (0 , 3 , 0) 2 (1 , 4 , 6) 4 (2 , 2 , 1) 4 3 1 2 (1 , 3 , 2) (0 , 3 , 0) (2 , 2 , 1) (1 , 4 , 6) (0 , 3 , 4) (0 , 3 , 4) 3 (1 , 3 , 2) N N Figure 3.5 The construction of N from N . The specification is the balance vector and ( l,u,c ). For clarity only one arc of N has a description of bounds and cost. See Figure 3.5 for an example of the construction. It is not difficult to show the following result. Lemma 3.2.4 Let N and N be as described above. Then every feasible flow in N corresponds to a feasible flow in N = ( V ( D ST ) ,A ( D ST ) ,l ,u ,b ,c ) and vice versa. Furthermore, the costs of these flows are the same. Proof: Exercise 3.6. ut 104 3. Flows in Networks 3.3 Flow Decompositions In this section we consider a network N = ( V,A,l ≡ ,u ) and denote by D = ( V,A ) the underlying digraph of N . By a path or cycle in N we mean a directed path or cycle in D . We will show that every flow in a network can be decomposed into a small number of very simple flows in the same network. Besides being a nice elementary mathematical result, this also has very important algorithmic consequences as will be clear from the succeeding sections. A path flow f ( P ) along a path P in N is a flow with the property that there is some number k ∈ R such that f ( P ) ij = k if ij is an arc of P and otherwise f ( P ) ij = 0. Analogously, we can define a cycle flow f ( W ) for any cycle W in D . The arc sum of two flows x,x , denoted x + x , is simply the flow obtained by adding the two flows arcwise. Theorem 3.3.1 Every flow x in N can be represented as the arc sum of some path and cycle flows f ( P 1 ) ,f ( P 2 ) ,...,f ( P α ) ,f ( C 1 ) ,...,f ( C β ) with the following two properties: (a) Every directed path P i , 1 ≤ i ≤ α with positive flow connects a source vertex to a sink vertex....
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This document was uploaded on 08/10/2011.
 Spring '11
 Algorithms

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