Georgia Inst. of Technology
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CS 8002
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This is an advanced topics course, intended for graduate students in theoretical computer science, mathematics and ACO program.
Announcement : Notes for first two lectures are now available (see below). Watch out for updates. Thanks to all hardworking writers !
Many optimization problems of theoretical and practical interest are NP-complete, meaning it is impossible to compute exact solutions to these problems in polynomial time unless P = NP. A natural way to cope with this curse of NP-completeness is to seek approximate solutions instead of exact solutions. An algorithm with approximation ratio C computes, for every problem instance, a solution whose cost is within a factor C of the optimum. Optimization problems exhibit a wide range of behavior in their approximability. It is well-known that Bin-Packing has an approximation algorithm with ratio 1+\epsilon for every \epsilon > 0. In theory jargon, we say that Bin-Packing has a polynomial-time approximation scheme (PTAS). However, it wasn't known till the early 90s whether problems like MAX-3SAT, Vertex Cover, and MAX-CUT have a PTAS. A celebrated result called the PCP Theorem finally showed that these problems have no PTAS unless P = NP. Such results that rule out the possibility of good approximation algorithms (under complexity theoretic assumptions like P != NP) are called inapproximability results or hardness of approximation results.
The PCP Theorem has an equivalent formulation from the point of view of proof checking. The PCP theorem states that every NP-statement has a probabilistically checkable proof, i.e. a proof which can be "spot-checked" by reading only a constant number of bits from the proof. These bits are selected by a randomized process using a very limited amount of randomness. The checking process always accepts a correct proof of a correct statement and rejects any cheating proof of an incorrect statement with high probability. The term "holographic proof" is sometimes used to highlight this feature that a cheating proof must be wrong everywhere and therefore, can be detected by a spot-check. The discovery of the connection between proof checking and inapproximability results is one of the most exciting theoretical developments in the last decade. Since then, PCPs have led to several breakthrough results in inapproximability theory, e.g. tight hardness results for Clique, MAX-3SAT, and Set Cover.
This course will cover many of the inapproximability results and PCPs used to prove them. No prior knowledge will be assumed, except the basic theory of NP-completeness. Participants are expected to scribe notes for one lecture and/or give one presentation. No assignments/exams !
Professor: Subhash Khot - 234 CoC, 404-385-6603 khot@cc.gatech.edu
Template latex files for scribe-notes can be found here (stolen from Sanjeev Arora's course at Princeton).Course Syllabus
Here is a tentative list of topics. The plan is to use the PCP Theorem as a black-box and focus on the hardness results. Proof of the PCP Theorem will be covered towards the end of the course.
Lecture Notes
References
PCP literature is extensive and often very technical. Here are good places to check out.