solu10 - This le contains the exercises, hints, and...

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This fle contains the exercises, hints, and solutions For Chapter 10 oF the book ”Introduction to the Design and Analysis oF Algorithms,” 2nd edition, by A. Levitin. The problems that might be challenging For at least some students are marked by ± ; those that might be difficult For a majority oF students are marked by ² . Exercises 10.1 1. Solve the Following linear programming problems geometrically. (a) maximize 3 x + y subject to x + y 1 2 x + y 4 x 0 ,y 0 (b) maximize x +2 y subject to 4 x y y 3+ x x 0 0 2. Consider the linear programming problem minimize c 1 x + c 2 y subject to x + y 4 x +3 y 6 x 0 0 where c 1 and c 2 aresomerea lnumbersnotbothequa ltozero . (a) Give an example oF the coefficient values c 1 and c 2 For which the problem has a unique optimal solution. (b) Give an example oF the coefficient values c 1 and c 2 For which the problem has infnitely many optimal solutions. (c) Give an example oF the coefficient values c 1 and c 2 For which the problem does not have an optimal solution. 3. Would the solution to problem (10.2) be di±erent iF its inequality con- straints were strict, i.e., x + y< 4 and x 6 , respectively? 4. Trace the simplex method on (a) the problem oF Exercise 1a. (b) the problem oF Exercise 1b. 5. Trace the simplex method on the problem oF Example 1 in Section 6.6 1
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(a) ± by hand. (b) by using one of the implementations available on the Internet. 6. Determine how many iterations the simplex method needs to solve the problem maximize n j =1 x j subject to 0 x j b j , where b j > 0 for j =1 , 2 ,...,n. 7. Can we apply the simplex method to solve the knapsack problem (see Example 2 in Section 6.6)? If you answer yes, indicate whether it is a good algorithm for the problem in question; if you answer no, explain why not. 8. ± Prove that no linear programming problem can have exactly k 1 optimal solutions unless k . 9. If a linear programming problem maximize n j =1 c j x j subject to n j =1 a ij x j b i for i , 2 ,...,m x 1 ,x 2 ,...,x n 0 is considered as primal ,thenits dual is deFned as the linear program- ming problem minimize m i =1 b i y i subject to m i =1 a ij y i c j for j , 2 ,...,n y 1 ,y 2 ,...,y m 0 . (a) Express the primal and dual problems in matrix notations. (b) ±ind the dual of the following linear programming problem maximize x 1 +4 x 2 x 3 subject to x 1 + x 2 + x 3 6 x 1 x 2 2 x 3 2 x 1 2 3 0 . (c) Solve the primal and dual problems and compare the optimal values of their objective functions. 10. ± Parliament pacifcation In a parliament, each parliamentarian has at most three enemies. Design an algorithm that divides the parliament into two chambers in such a way that no parliamentarian has more than one enemy in his or her chamber (after [Sav03], p.1, #4).
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solu10 - This le contains the exercises, hints, and...

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