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Unformatted text preview: Massachusetts Institute of Technology Handout 15 6.854J/18.415J: Advanced Algorithms Wednesday, October 28, 2009 David Karger Problem Set 8 Due: Friday, November 6, 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. In class we saw a greedy algorithm for set cover that achieved an approxima- tion ratio of O (log n ) given n sets. We can achieve the same ratio with a linear program- ming/randomized rounding solution. (a) Devise a linear programming relaxation for set cover, generalizing the one we saw in class for vertex cover. (b) Suppose we apply randomized rounding to the fractional set-usage variables. Argue that each element is covered with constant probability. (You will need to upper bound Q (1- x i ) when x i 1.) (c) By applying the rounding experiment repeatedly, arrange for every element to be covered with high probability while using only O (log n ) times the optimum number of sets.number of sets....
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This note was uploaded on 11/24/2009 for the course CS 6.854/18.4 taught by Professor Davidkarger during the Fall '09 term at MIT.
- Fall '09