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Unformatted text preview: Massachusetts Institute of Technology Handout 12 6.854J/18.415J: Advanced Algorithms Wed, October 21, 2009 David Karger Problem Set 5 Solutions Wed, October 21, 2009 Problem 1. For reference, the linear program is as follows: minimize cx s.t. x 1 + x 2 ≥ 1 x 1 + 2 x 2 ≤ 3 x 1 ≥ x 2 ≥ x 3 ≥ (a) For c = ( − 1 , , 0), this minimizes the quantity − x 1 , so it maximizes x 1 . We have x 1 + 2 x 2 ≤ 3 with x 2 ≥ 0, so the most x 1 can be is 3. This sets x 2 to 0, and all the other constraints are satisfied if x 3 ≥ 0. Thus we have the optimal value of3 and the set of possible solutions { (3 , , x 3 ) : x 3 ≥ } . (b) For c = (0 , 1 , 0), this minimizes the quantity x 2 . Given that x 2 ≥ 0, we set x 2 to 0. The rest of the constraints force 1 ≤ x 1 ≤ 3 and x 3 ≥ 0. Thus we have the optimal value of 0 and the set of possible solutions { ( x 1 , , x 3 ) : 1 ≤ x 1 ≤ 3 , x 3 ≥ } . (c) For c = (0 , , − 1), this minimizes the quantity x 3 , so it maximizes x 3 . Since the only constraint on x 3 is that it be nonnegative, the solution is unbounded and has an optimal value of −∞ . The set of possible solutions is { ( x 1 , x 2 , ∞ ) : x 1 + x 2 ≥ 1 , x 1 + 2 x 2 ≤ 3 , x 1 ≥ , x 2 ≥ } . Problem 2. (a) Let x i be the variable that denotes the amount of money we wish to trade with client i . We want to maximize the amount of yen we make, so we want to maximize the sum of the x i ’s that trade from anything to yen, which we denote by y , minus the amount of yen we trade away. Making sure we optimize for the final amount of yen is important because otherwise we would simply optimize for a maximum amount of yen we could achieve over time, which could mean trading away all the yen we got into pesos and trading it back into yen again, increasing the LP’s optimal value but not actually getting more yen. We denote by d the dollar currency. When we write i = ( a i , b i ), we mean the client i that trades currency a i for b i . We can write that as: max summationdisplay i =( a i ,y ) r i x i − summationdisplay i =( y,a i ) y 2 Handout 12: Problem Set 5 Solutions...
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 Fall '09
 DavidKarger
 Algorithms, Optimization, Constraints, LP, Howard Staunton, optimal value

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