week2 - 3 The maximum flow problem Although it is a special...

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3 The maximum fow problem Although it is a special case of the minimum-cost network Fow problem, the maximum fow problem is of great importance in its own right. This problem can be motivated by the following setting. Imagine that you have a network of pipes that are used to ship, for example, oil from its source, to where is it is re±ned. Each pipe in the network can maintain a certain capacity of Fow (per second), which depends on its cross-section, and other less signi±cant factors. At what rate can we deliver oil to the re±nery? The network of pipes corresponds to a directed graph G = ( N, E ). Each directed edge ( i, j ) E has a speci±ed capacity u ij . There is a speci±ed source node s N and a speci±ed sink node t N . This is the entire input to the maximum Fow problem. To specify a solution for this input, we must give a Fow value x ij for each directed edge ( i.j ) E . Such a solution x is feasible if 1. 0 x ij u ij for each ( i.j ) E ; and 2. X j :( j,i ) E x ji = X j :( i,j ) E x ij for each node i N - { s, t } (i.e., each node that is neither the source nor the sink). The ±rst type of constraints, the inequalities, are called capacity constraints and the second type, the equations, are called fow conservation constraints . (These are exactly the constraints of the LP formulation that we gave in the previous section.) The value of a Fow x is the total net Fow into t , which is equal to X j :( j,t ) E x jt - X k :( t,k ) E x tk . This is the objective function for the maximum Fow problem; in other words, we wish to ±nd a feasible Fow of maximum value. How big can the optimal Fow value be? Partition the vertices N into a set S containing the source s and a set T containing the sink t . (The sets S and T form a partition of N if each node in N is in exactly one of S and T .) We shall call such a partition ( S, T ) a cut ; note that the de±nition of a cut requires that s S and t T . Observe that every unit of Fow that goes from node s to node t must at some point pass along an edge ( i, j ) E where 19
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i S and j T . However, there are only so many directed edges of this form, and each such edge ( i, j ) has an upper bound u ij on the total Fow that can use it. If we let D ( S, T ) denote the set of directed edges ( i, j ) for which i S and j T , then the capacity of the cut ( S, T ) is equal to ( i,j ) D ( S,T ) u ij . Since each unit of Fow from s to t “uses up” one unit of the capacity of the cut ( S, T ), the value of the maximum Fow is at most the capacity of the cut. This claim is true for any cut ( S, T ). ±or the same input, some cuts may have large capacity, and some cuts may have small capacity. However, for each cut, its capacity places a limit on the value of the maximum Fow.
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This note was uploaded on 04/06/2008 for the course ORIE 321 taught by Professor Shmoys/lewis during the Spring '07 term at Cornell University (Engineering School).

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week2 - 3 The maximum flow problem Although it is a special...

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