1 Graph Powering Cont. PCP proof by Irit Dinur Presented by Israel Gerbi

2 Goal Reduction Goal: Input: Constraint graph (G=(V,E),C, ) where G is an (n,d, ) expander, and < d, Output: A new graph (G’,C’) with larger gap (denoted gap’), where If gap = 0, Otherwise.

3 Last Lecture G’ Construction V’ = V B=C·t C = const.

4 E’: How to generate an edge? Pick a random vertex a Take a step along a random edge out of the current vertex. Decide to stop with probability 1/t. Throw edge if above path has length>B We get weighted edges, why?

5 C’: New constraints

6 Plurality Assignment

7 New Plurality: Formal Definition

8 Last Week Analysis Definition: F is a subset of E which includes all edges that are not satisfied by σ. |F|/|E|≥gap We throw edges from F until |F|/|E|=min(gap,1/t)

9 Gap’ Analysis Reminder Lemma from last lesson Over all paths from a to b (weighted e’ edges)

10 To Work… Starting with more Definitions S := Total number of steps in our RW N F := Number of steps that were in F N F * := Number of steps that were in F, if our RW wasn’t limited to B steps

11 Motivation In more detail, we show:

12 Motivation cont. Second Moment Method says: We wanted to show:

13 Back to the Beginning We can now choose t so the new gap would be twice as large!

14 Expectation of N F The graph is d regular

15 Cutting Off the Tail We will now bound

16 The Tail

17 (1) Proof Combining the two results above we get: We finished proving (1). We now turn to (2)

18 (2) Proof We now show(2): Obviously,

19 (2) Proof Cont. Lemma from first lesson

20 (2) Proof Cont.

21 Second Moment Method Lemma (Second Moment Method): If X is a nonnegative r.v then Proof: Cauchy Schwartz inequality

22 Second Moment Method Proof We have: Arranging: Therefore: