# Expanders, Property Testing and the PCP theorem

## Lecture 3: Outline of Dinur’s Proof

Posted in lectures by HQN on September 5, 2008

In today’s lecture, I discussed the following:

1. A list of results regarding various PCPs following the outline in Madhu Sudan’s lecture notes (See Section 2.2. on History of Results).
2. Mentioned that the PCP theorem is equivalent to the NP-Hardness of Gap-MaxE3SAT$_{1,s}$ for some constant $s \in (0,1)$, which is equivalent to the NP-Hardness of approximating MaxE3SAT to a certain constant ratio $\alpha<1$.
3. Defined the so-called Gap-Max-CG-$\Sigma_{1,s}$ problem, whose NP-Hardness is also equivalent to the PCP theorem
4. Briefly showed one direction of the PCP theorem, namely PCP[O(log n), O(1)]$\subseteq$ NP
5. Outlined Dinur’s proof of the PCP theorem.

Another article of interest is from Radnakrishnan and Sudan.

On Monday, I will prove #2 above formally, and start talking about expanders.