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CSOR W4246 – Fall, 2014 Homework 4 – Theoretical part Out: Monday, October 27, 2014 Due: 6pm, Monday, November 10, 2014 This problem set is organized as follows: the first two problems reinforce class material and the next two problems are practical problems. As usual, keep your answers clear and concise, and make sure that your hand-writing is legible and that your name is clearly written on your homework. Collaboration is limited to discussion of ideas only: you should write up your solutions
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  CSOR W4246 – Fall, 2014 Homework 4 – Theoretical part Out: Monday, October 27, 2014Due: 6pm, Monday, November 10, 2014 This problem set is organized as follows: the first two problems reinforce class material and the next two problems are practical problems.As usual, keep your answers clear and concise, and make sure that your hand-writing is legible and that your name is clearly written on your homework. Collaboration is limited to discussion of ideas only: you should write up your solutions entirely on your own.For problem 3, you may use the LP solver of your choice: there are many options freely available.Please make sure you cite which one you used. 1. (20 points) For the following network, with edge capacities as shown, find the maximum flowfrom  s  to  t  along with a minimum cut. ! ! ! # $ # ! #$$%!& % $#& $ #$ 2. (20 points) There are many variations on the maximum flow problem. For the followingtwo natural generalizations, show how to solve the more general problem by  reducing  itto the srcinal max-flow problem (thereby showing that these problems also admit efficientsolutions). ã  There are multiple sources and multiple sinks, and we wish to maximize the flow betweenall sources and sinks. ã  Both the edges  and the vertices   (except for  s  and  t ) have capacities. The flow into andout of a vertex cannot exceed the capacity of the vertex.  3. (20 points) You are given the following data points in the plane(1 , 3) , (2 , 5) , (3 , 7) , (5 , 11) , (7 , 14) , (8 , 15) , (10 , 19)You want to find a line that approximately passes through these points (no line is a perfectfit). Write a linear program to find the line that minimizes the  maximum absolute error  max 1 ≤ i ≤ 7 | ax i  +  by i  −  c | and provide the optimal solution and its objective value.4. (25 points) You have a data set consisting of items that a consumer has purchased on differentdays and the prices at which they were purchased. You are tasked with finding bounds onthe consumer’s values (i.e., willingness to pay) for different items.Specifically, let the set of items be  V    =  { 1 ,...,n } . Let  t  = 1 ,...,k  index the days, and let  p t 1 ,...,p tn  be the prices for the different items on day  t . If on day  t  the consumer purchased,say, item 1, this means that it was the most preferred item at the prices on that day, leadingto the inequalities: v 1  −  p t 1  ≥  v i  −  p ti  ( i  ∈  V    ) . Let  T  i  be the indices of the days on which item  i  was purchased, as recorded in your dataset. You would like to compute  upper bounds   on the consumer’s values. (a) (7 points)  Let  w ij  = min t ∈ T  i {  p t j  −  p ti } . Formulate the system of inequalities that con-strains the values in terms of the variables  v i  and the constants  w ij . (b) (5 points)  Show that if   v  = ( v  1 ,...,v  n ) and  v  = ( v  1 ,...,v  n ) are feasible solutions tothe constraints, then so is v  ∨  v  = (max { v  1 ,v  1 } ,..., max { v  n ,v  n } ) . From this result, you see that if you maximize the objective   i ∈ V    v i  subject to thefeasibility constraints, the solution will give upper bounds on the values. (c) (13 points)  Write out the linear program you have derived so far for the value bounds,and provide its dual. (d)  Extra credit (15 points)  You notice that your primal LP looks very similar to themin-cut LP we saw in class. What problem is the dual solving in your case? Hint: Note that if   ( v  1 ,v  2 ,...,v  n )  is a feasible solution, then so is   ( v  1 + c,v  2 + c,...,v  n + c )  for a constant   c . Therefore, in the primal you can also include the constraint   v 1  = 0 .
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