Wednesday, May 2, 2007

Post Comments on PCPs, Average-Case Complexity Here

Dear 6.841 students,

As part of your course participation requirement, please post/send in
your comments on the material covered in the 3 lectures on PCPs and the 2
lectures on Average-case complexity.

Here are the instructions for sending in your comments:

The idea is to send a brief overview *in your own words*
of what you thought was covered. Follow this up with a few words
saying what you liked and what you did not like. Any other comments,
questions, topics to be clarified would also be welcome. Your
comments could range from anywhere from a couple of paras to a whole
volume! Post your comments on the blog at http://mit6841.blogspot.com/
(or email them to Swastik).

63 comments:

David Glasser said...
This comment has been removed by the author.
David Glasser said...

^PCPs: the basic idea here is that there exists a format for certificates for any language in NP such that a probabilistic verifier only needs to look at a very small constant number of bits of the proof to be convinced that a word is in the language. This is pretty amazing; for example, this is like saying that we could believe that a graph is three-colorable by only asking about the coloring of a couple vertices, when it's possible for a graph to not be three-colorable because only one edge is inconsistent (and thus there is only a 1/m chance of discovering that such a "proof" is faked). And in fact, Dinur's proof of the PCP Theorem is based on a generalization of graph coloring which begins by converting any NP problem into such a 3-colorable graph; the main body of Dinur's proof consists of analyzing a few transformations of the graph coloring problem which enlarge the "gap" in the error probability until it passes a constant.

Average-case complexity: One criticism of the theory of NP-completeness is that while we may claim that languages like SAT, 3-coloring, and the travelling salesman problem are "as hard" as any language in NP, and are unlikely to be solvable in polynomial time, this is purely a worst-case analysis, and there can still exist algorithms which work efficiently on many real cases of the problem. Average-case complexity augments a decision problem with a probability distribution over its inputs; an algorithm is considered to be average-case efficient if there is a polynomial P such that the probability (when choosing inputs from the given distribution) that the algorithm takes longer than P to compute is negligible. The open problem which is equivalent to P vs NP in this context is DNP vs Average-BPP. We also described DNP-complete problems, which require mapping distributions as well as the problem itself.

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