Princeton University
Computer Science Department

Mini-course on Projection PCPs – Spring 2009

Dana Moshkovitz




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.


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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)





March 26


Introduction: projection games and unique games, connection to CSPs, meaning as PCPs and applications to hardness of approximation


March 27


No class; Intractability Center Meeting


April 2


No class; FOCS’09 deadline


April 3


Example for application to hardness of approximation: Max-Cut (for unique games); the long-code and dictator testing


April 9


The Max-Cut example continued and (hints to) the generalization by Prasad Raghavendra


April 10


Low Degree Polynomials - The Coding-Theoretic Perspective: The Reed-Muller code (distance, rate, recursive structure), local decoding


April 16


Local Testing of Low Degree Polynomials: The Raz-Safra Plane vs. Point and Plane vs. Plane testers


April 17


Analysis of The Plane vs. Plane tester for the three-dimensional case continued


April 23


Algebraic construction of PCP with large alphabet: The Zero Testing Problem and its gap version, the [LFKN] Sum-Check


April 24

Algebraic construction of PCP with large alphabet: The Manifold vs. Point construction




April 30


Combinatorial transformations on projection games: right degree reduction


May 1st


No class; Intractability Center Meeting


May 7


Analogy to codes and code concatenation, right alphabet reduction; switching sides


May 8


Composition for projection games and completing the proof of the Projection Games Theorem