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Algorithms_Part13

Algorithms_Part13 - S Dasgupta C.H Papadimitriou and U.V...

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S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 241 7.12. For the linear program max x 1 - 2 x 3 x 1 - x 2 1 2 x 2 - x 3 1 x 1 , x 2 , x 3 0 prove that the solution ( x 1 , x 2 , x 3 ) = (3 / 2 , 1 / 2 , 0) is optimal. 7.13. Matching pennies. In this simple two-player game, the players (call them R and C ) each choose an outcome, heads or tails . If both outcomes are equal, C gives a dollar to R ; if the outcomes are different, R gives a dollar to C . (a) Represent the payoffs by a 2 × 2 matrix. (b) What is the value of this game, and what are the optimal strategies for the two players? 7.14. The pizza business in Little Town is split between two rivals, Tony and Joey. They are each investigating strategies to steal business away from the other. Joey is considering either lowering prices or cutting bigger slices. Tony is looking into starting up a line of gourmet pizzas, or offering outdoor seating, or giving free sodas at lunchtime. The effects of these various strategies are summarized in the following payoff matrix (entries are dozens of pizzas, Joey’s gain and Tony’s loss). T ONY Gourmet Seating Free soda J OEY Lower price +2 0 - 3 Bigger slices - 1 - 2 +1 For instance, if Joey reduces prices and Tony goes with the gourmet option, then Tony will lose 2 dozen pizzas worth of business to Joey. What is the value of this game, and what are the optimal strategies for Tony and Joey? 7.15. Find the value of the game specified by the following payoff matrix. 0 0 - 1 - 1 0 1 - 2 - 1 - 1 - 1 1 1 - 1 0 0 1 1 - 2 0 - 3 1 - 1 - 1 - 1 0 - 3 2 - 1 0 - 2 1 - 1 ( Hint: Consider the mixed strategies (1 / 3 , 0 , 0 , 1 / 2 , 1 / 6 , 0 , 0 , 0) and (2 / 3 , 0 , 0 , 1 / 3) .) 7.16. A salad is any combination of the following ingredients: (1) tomato, (2) lettuce, (3) spinach, (4) carrot, and (5) oil. Each salad must contain: (A) at least 15 grams of protein, (B) at least 2 and at most 6 grams of fat, (C) at least 4 grams of carbohydrates, (D) at most 100 milligrams of sodium. Furthermore, (E) you do not want your salad to be more than 50% greens by mass. The nutritional contents of these ingredients (per 100 grams) are

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242 Algorithms ingredient energy protein fat carbohydrate sodium (kcal) (grams) (grams) (grams) (milligrams) tomato 21 0.85 0.33 4.64 9.00 lettuce 16 1.62 0.20 2.37 8.00 spinach 371 12.78 1.58 74.69 7.00 carrot 346 8.39 1.39 80.70 508.20 oil 884 0.00 100.00 0.00 0.00 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. 7.17. Consider the following network (the numbers are edge capacities). A B C D T S 7 6 3 4 2 2 5 9 (a) Find the maximum flow f and a minimum cut. (b) Draw the residual graph G f (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 flow. List all bottleneck edges in the above network.
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