ps6 - Massachusetts Institute of Technology Handout 11...

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Unformatted text preview: Massachusetts Institute of Technology Handout 11 6.854J/18.415J: Advanced Algorithms Wednesday, October 14, 2009 David Karger Problem Set 6 Due: Wednesday, October 21, 2009. Collaboration policy: collaboration is strongly encouraged . However, remember that 1. You must write up your own solutions, independently. 2. You must record the name of every collaborator. 3. You must actually participate in solving all the problems. This is difficult in very large groups, so you should keep your collaboration groups limited to 3 or 4 people in a given week. 4. No bibles. This includes solutions posted to problems in previous years. Problem 1. Although the dual can tell you a lot about the structure of a problem, knowing an optimal dual solution does not in general help you solve the primal problem. Suppose we had an algorithm that could optimize an LP with an m n constraint matrix in O (( m + n ) k ) time given an optimal solution to the dual LP. (a) Argue that any LP optimization problem can be transformed into the following form: minimize x subject to Ax = b x (This LP has optimum value 0 if it is feasible, and if it is infeasible.) (b) What is the dual of this linear program? (c) Argue that, if the primal is feasible, then the dual has an obvious optimum solution. (d) Deduce that, given the hypothetical algorithm above, you can build an LP al- gorithm that will solve any LP without knowing a dual solution, in the same asymptotic time bounds as the algorithm above. Problem 2. Consider a graph in which edges have costs (possibly negative, representing profits). Suppose you want to find a minimum mean cycle in this graph: one with the minimum ratio of cost to length (number of edges). Going around such a cycle repeatedly (assuming it is negative) provides you with the maximum possible profit per unit length/time, so it is the fastest way to earn money if you are, for example, a delivery service. Finding a 2 Handout 11: Problem Set 6 minimum mean cycle is also an essential step in certain strongly polynomial min cost flow algorithms. Consider the following linear program: w = min summationdisplay c ij f ij summationdisplay j f ij- f ji = 0 ( i ) summationdisplay...
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ps6 - Massachusetts Institute of Technology Handout 11...

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