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The importance of linear programming stems from the astounding variety of problems that reduce to it and thereby bear witness to its expressive power In a sense, this next one is the ultimate application We are given a Boolean circuit, that is, a dag of gates of the following types Input gates have indegree zero, with value true or false

14 In an RC circuit, as the ratio XC /R approaches zero, the phase angle (a) approaches 90 (b) approaches 0 (c) stays the same (d) cannot be found 15 In a purely resistive circuit, the phase angle is (a) increasing (b) decreasing (c) 0 (d) 90 16 If XC /R = 1, then what is the phase angle (a) 0 (b) 45 (c) 90 (d) Impossible to find because there s not enough data given 17 In Fig 14-13, the impedance shown is (a) 802 + j323 (b) 323 + j802 (c) 802 j323 (d) 323 j802

Add 1 to the input Figure 14-11 shows some points plotted in Cartesian coordinates, along with the straight line that connects them, for the following function:

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The CIRCUIT VALUE problem is the following: when the laws of Boolean logic are applied to the gates in topological order, does the output evaluate to true There is a simple, automatic way of translating this problem into a linear program Create a variable xg for each gate g, with constraints 0 x g 1 Add additional constraints for each type of gate:

y=x+1 These points lie along a straight line Note the similarity between this graph and the one shown in Fig 14-4 The only difference here is that the line is exactly 1 unit lower on the coordinate plane

18 In Fig 14-13, note that the R and X C scale divisions are not the same size What is the actual phase angle (a) 142 (b) About 60 , from the looks of it (c) 589 (d) 886 19 Suppose an RC circuit consists of a 150-pF capacitor and a 330- resistor in series What is the phase angle at a frequency of 134 MHz (a) 674 (b) 226 (c) 244 (d) 656 20 Suppose an RC circuit has a capacitance of 0015 F The resistance is 52 What is the phase angle at 90 kHz (a) 24 (b) 0017 (c) 66 (d) None of the above

Square the input In Fig 14-12, several points are plotted, and the curve connecting them is drawn, for the function

These constraints force all the gates to take on exactly the right values 0 for false, and 1 for true We don t need to maximize or minimize anything, and we can read the answer off from the variable xo corresponding to the output gate This is a straightforward reduction to linear programming, from a problem that may not seem very interesting at rst However, the CIRCUIT VALUE problem is in a sense the most general problem solvable in polynomial time! After all, any algorithm will eventually run on a computer, and the computer is ultimately a Boolean combinational circuit implemented on a chip If the algorithm runs in polynomial time, it can be rendered as a Boolean circuit consisting of polynomially many copies of the computer s circuit, one per unit of time, with the values of the gates in one layer used to compute the values for the next Hence, the fact that CIRCUIT VALUE reduces to linear programming means that all problems that can be solved in polynomial time do! In our next topic, NP-completeness, we shall see that many hard problems reduce, much the same way, to integer programming, linear programming s dif cult twin 218

w = v2 This graph is the same curve as the one shown in Fig 14-9, but the variable names are different

also get acquainted with admittance, the extent to which an ac circuit allows (or admits) current flow, rather than impeding it As we develop these concepts, let s review, and then expand on, some of the material presented in the previous couple of chapters

Cube the input Figure 14-13 shows what happens when the independent variable is cubed rather than squared Several points, along with the curve connecting them, are plotted for the function

Another parting thought: by what other means can the circuit evaluation problem be solved Let s think a circuit is a dag And what algorithmic technique is most appropriate for solving problems on dags That s right: dynamic programming! Together with linear programming, the world s two most general algorithmic techniques

y 6 (4,5) 4 2 (0,1) x 6 2 ( 4, 3) 4 6 2 4 6

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