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What if we can t win If we find it impossible to get a matrix into the unit diagonal form, our failure indicates one of four things:

Find a linear programming applet on the Web and use it to make the salad with the fewest calories under the nutritional constraints Describe your linear programming formulation and the optimal solution (the quantity of each ingredient and the value) Cite the Web resources that you used 717 Consider the following network (the numbers are edge capacities)

We didn t try hard enough We made a mistake somewhere The original system is inconsistent The original system is redundant

The system to be solved Now let s methodically tackle a three-by-three linear system and solve it using the matrix morphing game Here are the equations:

Zpri/Zsec = (Tpri/Tsec )2 Zpri/Zsec = (Epri/Esec )2 The inverses of these formulas, in which the turns ratio or voltage-transfer ratio are expressed in terms of the impedance-transfer ratio, are: Tpri/Tsec = (Zpri/Zsec)1/2 Epri/Esec = (Zpri/Zsec)1/2

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(a) Find the maximum ow f and a minimum cut (b) Draw the residual graph Gf (along with its edge capacities) In this residual network, mark the vertices reachable from S and the vertices from which T is reachable (c) An edge of a network is called a bottleneck edge if increasing its capacity results in an increase in the maximum ow List all bottleneck edges in the above network (d) Give a very simple example (containing at most four nodes) of a network which has no bottleneck edges (e) Give an ef cient algorithm to identify all bottleneck edges in a network (Hint: Start by running the usual network ow algorithm, and then examine the residual graph) 718 There are many common variations of the maximum ow problem Here are four of them (a) There are many sources and many sinks, and we wish to maximize the total ow from all sources to all sinks (b) Each vertex also has a capacity on the maximum ow that can enter it (c) Each edge has not only a capacity, but also a lower bound on the ow it must carry (d) The outgoing ow from each node u is not the same as the incoming ow, but is smaller by a factor of (1 u ), where u is a loss coef cient associated with node u

Formatting the equations None of these equations is in the proper form for assembling a matrix Let s get them that way! Fortunately, the maneuvers are simple With the first equation, we can subtract 2y from each side to get

Each of these can be solved ef ciently Show this by reducing (a) and (b) to the original max- ow problem, and reducing (c) and (d) to linear programming 719 Suppose someone presents you with a solution to a max- ow problem on some network Give a linear time algorithm to determine whether the solution does indeed give a maximum ow 720 Consider the following generalization of the maximum ow problem You are given a directed network G = (V, E) with edge capacities {ce } Instead of a single (s, t) pair, you are given multiple pairs (s1 , t1 ), (s2 , t2 ), , (sk , tk ), where the si are sources of G and the ti are sinks of G You are also given k demands d1 , , dk The goal is to nd k ows f (1) , , f (k) with the following properties:

3x 2y + z = 11 With the second equation, we can subtract x from each side to obtain x + 4y + 2z = 0 With the third equation, we can subtract 3z from each side, getting 5x + y 3z = 20 We now have the equations in form, and can state the whole system like this: 3x 2y + z = 11 x + 4y + 2z = 0 5x + y 3z = 20

Problem 18-3 Consider a situation in which a transformer is needed to match an input impedance of 500 , purely resistive, to an output impedance of 300 , also purely resistive What is the required turns ratio Tpri/Tsec The required transformer will have a step-up impedance ratio of Zpri/Zsec = 500/300 = 1/600 From the preceding formulas:

Building the matrix To construct the matrix, we remove all the variables and arrange the remaining numbers into an orderly array, paying close attention to the signs:

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