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ps9 - Massachusetts Institute of Technology 6.854J/18.415J...

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Massachusetts Institute of Technology Handout 17 6.854J/18.415J: Advanced Algorithms Friday, November 6, 2009 David Karger Problem Set 9 Due: Friday, November 13, 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. Treewidth. (a) In class we showed how the decision version of SAT is fixed-parameter tractable with respect to the treewidth. Show that maximum satisfiability , the problem of simultaneously satisfying as many clauses as possible, is also FPT with respect to treewidth. Your should need only a short paragraph to explain the change to the decidability algorithm and why it works. (b) Consider the (nonparameterized) vertex cover problem: given a graph G , find a minimum-size set of vertices that covers all edges (i.e., every edge has at least one endpoint in the vertex cover). Show that vertex cover is fixed-parameter tractable with respect to the treewidth parameter. Problem 2. Here we will generalize from the Double Coverage algorithm for k-server on the line to k-server on trees. We say that a server s i
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