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Princeton University
Computer Science Department
Mini-course on
Projection PCPs – Spring 2009
Dana Moshkovitz
Summary
The Probabilistically Checkable Proofs (PCP) Theorem (Arora-Safra, Arora-Lund-Motwani-Szegedy-Sudan, 1992) states that every mathematical proof can be written in a format that allows probabilistic checking by making only a constant number of queries to the proof. The PCP theorem – not only extremely interesting by itself – was a breakthrough in proving NP-hardness of approximation problems.
A Projection PCP is a special kind of PCP, in which the proof is partitioned into two parts, A and B, and the verifier performs one query to each part, such that the answer to the A query determines at most one satisfying answer to the B query.
Projection PCPs are a starting point for a host of hardness of approximation results, such as those of 3SAT, 3LIN, Set-Cover, etc.
In the mini-course we will present a recent construction of a projection PCP with error that tends to 0 [M-Raz, 2008] with a beautiful formulation by [Dinur-Harsha, 2009] that generalizes and simplifies the original construction.
In the process of showing the construction, we will cover algebraic PCP techniques that are of independent interest, such as sum check, low degree testing and local testing/decoding of polynomials. We will also discuss hardness of approximation and the motivations arising from there.
The mini-course will complement the PCP part of Boaz’s course, but will be entirely self-contained.
Time & Place
The course will be on Thursdays and Fridays, 10:30 – 12:00, at CS-402.
Mailing List
Subscribe here:
https://lists.cs.princeton.edu/mailman/listinfo/projection-pcp-mini-course
Lecture Notes
We’ll use the
style files for Sanjeev’s class. A macros.tex file
adapted to the mini-course is available.
Additional Reading
1.
Unique Games Hardness for
Max-Cut (Paper by Subhash Khot, Guy Kindler, Elchanan
Mossel, Ryan O’Donnell)
2. Unique Games Hardness for CSPs from Integrality Gaps for SDP (Paper by Prasad Raghavendra)
3. The Plane vs. Point Tester (The proof for the three-dimensional case of the Plane vs. Plane Tester is in the paper by Ran Raz and Muli Safra; full analysis of the Plane vs. Point Tester can be found here).
4. PCP Construction with Two Queries and Error tending to 0 (Paper with Ran Raz; read Chapters 1-3)
Plan
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Thursday, March 26 |
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Friday, March 27 |
No class; Intractability Center Meeting |
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Thursday, April 2 |
No class; FOCS’09 deadline |
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Friday, April 3 |
Example for application to hardness of approximation: Max-Cut (for unique games); the long-code and dictator testing |
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Thursday, April 9 |
The
Max-Cut example continued and (hints to) the generalization by Prasad
Raghavendra |
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Friday, April 10 |
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Thursday, April 16 |
Local
Testing of Low Degree Polynomials: The Raz-Safra Plane vs. Point and Plane
vs. Plane testers |
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Friday, April 17 |
Analysis of The Plane vs. Plane tester for the three-dimensional case continued |
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Thursday, April 23
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Algebraic construction of PCP with large alphabet: The Zero Testing Problem and its gap version, the [LFKN] Sum-Check |
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Friday, April 24 |
Algebraic construction of PCP with large alphabet: The Manifold vs. Point construction |
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Thursday, April 30 |
Combinatorial transformations on projection games: right
degree reduction |
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Friday, May 1st |
No class; Intractability Center Meeting |
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Thursday, May 7 |
Analogy to codes and code concatenation, right alphabet reduction; switching sides |
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Friday, May 8 |
Composition for projection games and completing the proof of the Projection Games Theorem |