# Notes on the PCP Theorem and the Hardness of Approximation: Part 1

In this note, we are going to state the PCP theorem and its relation to the hardness of approximating some NP-hard problem.

## PCP Theorem: the Interactive Proof View

Intuitively, a PCP (Probabilistically Checkable Proof) system is an interactive proof system where the verifier is given $r(n)$ random bits and he is allowed to look into the proof $\pi$ in $q(n)$ many locations. If the string $x \in \{0,1\}^n$ is indeed in the language, then there exists a proof so that the verifier always accepts. However, if $x$ is not in the language, no prover can convince this verier with probability more than $1/2$. The proof has to be short i.e., of size at most $q(n) 2^{r(n)}$. This class of language is designated as PCP[r(n), q(n)].

Theorem A (PCP theorem). Every NP language has a highly efficient PCP verifier. In particular,

$NP = PCP[O(\log n), O(1)]$.

# Forkable Strings are Rare

In a blockchain protocol such as Bitcoin, the users see the world as a sequence of states. A simple yet functional view of this world, for the purpose of analysis, is a Boolean string $w = w_1, w_2, \cdots$ of zeros and ones, where each bit is independently biased towards $1$ favoring the “bad guys.”

A bad guy is activated when $w_t = 1$ for some $t$. He may try to present the good guys with a conflicting view of the world, such as presenting multiple candidate blockchains of equal length. This view is called a “fork”. A string $w$ that allows the bad guy to fork (with nonnegligible probability) is called a “forkable string”. Naturally, we would like to show that forkable strings are rare: that the manipulative power of the bad guys over the good guys is negligible.

Claim ([1], Bound 2). Suppose $w =w_1, \cdots, w_n$ is a Boolean string, with every bit independently set to $1$ with probability $(1-\epsilon)/2$ for some $\epsilon < 1$. The probability that $w$ is forkable is at most $\exp(-\epsilon^3n/2)$.

In this post, we present a commentary on the proof that forkable strings are rare. I like the proof because it uses simple facts about random walks, generating functions, and stochastic domination to bound an apparently difficult random process.

# Spectral Sparsification by Spielman and Srivastava: How They Connected the Dots

In this post, I will discuss my own understanding of the spectral sparsification paper by Daniel Spielman and Nikhil Srivastava (STOC 2008). I will assume the following:

1. The reader is a beginner, like me, and have already glanced through the Spielman-Srivastava paper (from now on, the SS paper).
2. The reader has, like me, a basic understanding of spectral sparsification and associated concepts of matrix analysis. I will assume that she has read and understood the Section 2 (Preliminaries) of the SS paper.
3. The reader holds a copy of the SS paper while reading my post.

First, I will mention the main theorems (actually, I will mention only what they “roughly” say).