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lect23 - 6.841 Advanced Complexity Theory May 4, 2009...

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Unformatted text preview: 6.841 Advanced Complexity Theory May 4, 2009 Lecture 23 Lecturer: Madhu Sudan Scribe: Rishi Gupta Project Presentations Presentations are this Wednesday 9am-12pm and Thursday 9am-1pm. Slots may be moved around to space them out/ensure we dont run up against room reservation deadlines. The 20 minute limit will be enforced. Powerpoint is highly recommended, since you wont be able to write that much or get through much material in a 20 minute interactive session. If you do write on the board, you should have all your intermediate steps planned out beforehand. Also, know what youre going to do if we do make it interactive and you have a few minutes less than you think you do. Send in your paper and presentation before you present, preferably at least 24 hours before. Average-Case Complexity Today were going to give an overview of Average-Case Complexity . This was originally a three day lecture, and now its a one day lecture; whats being lost is the proofs. We will give motivation for the field, some definitions, and a flavor of the results. Motivation Weve already seen that worst-case instances of Permanent reduce to random instances of Permanent (Lip- ton), implying that for Permanent the average-case complexity equals the worst-case complexity. This is not very encouraging. However, we hear about SAT-solvers and other algorithms for NP-hard problems all the time. Some- times people are just mistaken (eg. they dont realize their graphs have some hidden constraint), sometimes they are just using an approximation algorithm, but sometimes it legitimately seems like the empirical, real-life problems are simply not the worst-case ones. The key is, the distribution of real-life problems is not the same as the distribution of random problems (say, of Clique). People are very interested in finding algorithms that work on real-life distributions of NP. An example might be compiler optimization. Another application of average-case complexity is cryptography. People would like to find functions f such that f is easy to compute and f- 1 is hard to compute (where easy/hard refer to polynomial-time, as usual). More strongly, people want functions f for which they both know the value of f- 1 on some inputs x i and where its still hard to compute f- 1 on those x i . This means its not just sufficient to solve all the average-case or easy instances of a function and declare rest as hard....
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lect23 - 6.841 Advanced Complexity Theory May 4, 2009...

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