

The Probabilistically Checkable Proofs (PCP) Theorem (AroraSafra, AroraLundMotwaniSzegedySudan, 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 NPhardness 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, SetCover, etc.
In the minicourse we will present a recent construction of a projection PCP with error that tends to 0 [MRaz, 2008] with a beautiful formulation by [DinurHarsha, 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 minicourse will complement the PCP part of Boaz’s course, but will be entirely selfcontained.
The course will be on Thursdays and Fridays, 10:30 – 12:00, at CS402.
Subscribe here:
https://lists.cs.princeton.edu/mailman/listinfo/projectionpcpminicourse
We’ll use the
style files for Sanjeev’s class. A macros.tex file
adapted to the minicourse is available.
1.
Unique Games Hardness for
MaxCut (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 threedimensional 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 13)
Thursday, March 26 

Friday, March 27 
No class; Intractability Center Meeting 
Thursday, April 2 
No class; FOCS’09 deadline 
Friday, April 3 
Example for application to hardness of approximation: MaxCut (for unique games); the longcode and dictator testing 
Thursday, April 9 
The
MaxCut example continued and (hints to) the generalization by Prasad
Raghavendra 
Friday, April 10 

Thursday, April 16 
Local
Testing of Low Degree Polynomials: The RazSafra Plane vs. Point and Plane
vs. Plane testers 
Friday, April 17 
Analysis of The Plane vs. Plane tester for the threedimensional case continued 
Thursday, April 23

Algebraic construction of PCP with large alphabet: The Zero Testing Problem and its gap version, the [LFKN] SumCheck 
Friday, April 24 
Algebraic construction of PCP with large alphabet: The Manifold vs. Point construction 
Thursday, April 30 
Combinatorial transformations on projection games: right
degree reduction 
Friday, May 1st 
No class; Intractability Center Meeting 
Thursday, May 7 
Analogy to codes and code concatenation, right alphabet reduction; switching sides 
Friday, May 8 
Composition for projection games and completing the proof of the Projection Games Theorem 