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Lecture 9 Dinur’s Proof of the PCP Theorem
CS369E: Expanders May 31, 2005
Lecture 9: Dinur’s Proof of the PCP Theorem
Lecturer: Prahladh Harsha Scribe: Krishnaram Kenthapadi
In this lecture, we will describe a recent and remarkable proof of the PCP theorem,
due to Irit Dinur [Din]. This proof is a beautiful application of expanders for “soundness
amplification” of randomized algorithms without using too many random bits.
Before describing the proof, we will first look at the PCP theorem, by relating it with
the theory of NP-completeness.
9.1 Hardness of Optimization Problems
The theory of NP-completeness, as developed by Cook, Levin, and Karp, states that any
language, L in NP is reducible to the Boolean satisfiability problem, 3SAT. By this, we
mean that for every instance, x of the language L, we can obtain a satisfiability instance, φ
such that x ∈ L if and only if φ is satisfiable. Thus, 3SAT is at least as hard as any other
problem in NP. Karp further showed that 3SAT can be reduced to other problems such as
CLIQUE and 3-COLORABILITY and hence that these problems are at least as hard as
any problem in NP. In other words, solving these problems optimally is as hard as solving
any other problem in NP optimally.
However the question of the hardness of approximation was left open. For instance,
can the following be true – finding a satisfying assignment for 3SAT is NP-hard, however
it is easy to find an assignment that satisfies 99% of the clauses. Questions such as Other
examples: can we approximate the clique size in a graph? Or, can we obtain a 3-coloring that
satisfies 99% of the edge constraints? In other words, is the approximation version of some
of NP-hard problems easier than the optimization versions. The PCP Theorem [FGLSS,
AS, ALMSS]
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