\documentclass{article}
\usepackage{amsmath}
\input{preamble.tex}
\setlength{\oddsidemargin}{.25in}
\setlength{\evensidemargin}{.25in}
\setlength{\textwidth}{6in}
\setlength{\topmargin}{-0.4in}
\setlength{\textheight}{9.5in}
 
\begin{document}
 
\lecture{17}{April 12, 2001}{Dan Spielman}{Ronnie Misra}
 
\section*{Recap}
Last time, we discussed the class of ``Arthur/Merln games''. Recall that:

\begin{definition}
L $\in$ AM(k(n)) if $\exists$ a PTIME verifier A and a polynomial p(n) s.t. \newline
\indent x $\in$ L $\Longrightarrow$ $\exists$ a p(n)-prover P s.t. Pr[(A $\leftarrow \rightarrow$ P) accepts] $>$ $\frac{2}{3}$ \newline
\indent x $\notin$ L $\Longrightarrow \forall$ p(n)-provers P, Pr[(A $\leftarrow \rightarrow$ P) accepts] $<$ $\frac{1}{3}$
\end{definition}

In this lecture, we will show some applications of IP, and discuss the class PCP.

\section*{Graph Isomorphism}
Although there are no known PTIME algorithms for graph isomorphism, we can show that ISO is probably not NP-complete:

\begin{theorem}
If ISO is NP-complete, then $\Sigma_3^P$ = $\Pi_3^P$.
\end{theorem}

\begin{proof}
In the problem set, we showed that MA $\subseteq$ AM
(proved as NP $\cdot$ BP $\cdot$ P $\subseteq$ BP $\cdot$ NP $\cdot$ P).
In fact, the AM speedup theorem says that: \newline
\indent AM(k(n)) $\subseteq$ AM($\frac{k(n)}{2}$ + 1) \newline
\indent $\Longrightarrow$ AM(k) $\subseteq$ AM $\forall$ constants k \newline
\indent $\Longrightarrow$ MAM $\subseteq$ AM \newline
We also showed in the problem set that AM ( = BP $\cdot$ NP $\cdot$ P ) $\subseteq$ NP/poly. \newline

\indent ISO is NP-complete \newline
\indent $\Longrightarrow$ NISO is coNP-complete \newline
\indent $\Longrightarrow$ coNP $\subseteq$ MAM \newline
\indent $\Longrightarrow$ coNP $\subseteq$ AM \newline
\indent $\Longrightarrow$ coNP $\subseteq$ NP/poly \newline
\indent $\Longrightarrow$ $\Sigma_3^P$ = $\Pi_3^P$.

\end{proof}

\section*{Probabilistically checkable proofs}

We would like to describe a ``tough verifier'' that accepts a proof that can be ``spot checked''.

\begin{definition}
A (R(n), Q(n)) verifier is a probabilistic, PTIME oracle TM $M^P$ that
\begin{itemize}
\item gets R(n) random bits
\item is limited to Q(n) bits from its oracle P
\end{itemize}
\end{definition}

\begin{definition}
L $\in$ PCP(R(n), Q(n)) if $\exists$ a (R(n), Q(n)) verifier V s.t. \newline
\indent x $\in$ L $\Longrightarrow$ $\exists$ $\Pi$ s.t. Pr[$V^\Pi$(x) accepts] = 1 \newline
\indent x $\notin$ L $\Longrightarrow \forall$ $\Pi$, Pr[$V^\Pi$(x) accepts] $<$ $\frac{1}{2}$ \newline
\end{definition}

Here, $\Pi$ is the probabilistically checkable proof (PCP). Note that we don't really think of $\Pi$ as an oracle. Instead, we are using OTM notation just as a way to show that V gets access to random bits of $\Pi$.

\begin{theorem}
NEXP = PCP(Poly(n), Poly(n))
\end{theorem}

Even though proofs for languages in NEXP are exponentially long, randomized access allows such proofs to be verified in PTIME.

\begin{theorem}
NP = PCP(O(log n), O(1))
\end{theorem}

As a consequence, we can use PCP to show that some approximation problems are NP-hard.

\section*{3SAT approximation is NP-hard}

\indent 3SAT $\subseteq$ NP = PCP(O(log n), O(1)) \newline
\indent $\Longrightarrow$ $\exists$ a (O(log n), O(1)) verifier for 3SAT.

Assume V is non-adaptive, or that it flips R = O(log n) coins, then queries the proof at Q = O(1) places determined only by these random bits. (If V is adaptive, we can construct a non-adaptive version; this version may use exponentially more queries, but this is still a constant.)

On input $\Phi$, assume V gets random bits r $\in$ $\{0, 1\}^R$. Let $q_{r,1}$ \ldots $q_{r,Q}$ denote the indices of the Q bits of $\Pi$ that V queries. Thus, if $\Pi$ = $\Pi_1$$\Pi_2$\ldots$\Pi_N$, V reads $\Pi_{q_{r,1}}$ \ldots $\Pi_{q_{r,Q}}$.

For each r $\in$ $\{0, 1\}^R$, construct a small 3SAT instance $\phi_r$ on inputs $\Pi_{q_{r,1}}$ \ldots $\Pi_{q_{r,Q}}$ and some auxilliary bits. Assume $\phi_r$ has at most S clauses. Since Q = O(1), S = O(1). $\phi_r$ should be satisfied when $\Pi_{q_{r,1}}$ \ldots $\Pi_{q_{r,Q}}$ have values that would cause V to accept.

Let $\Phi^\prime$ = $\underset{r \in \{0,1\}^R}{\Lambda} \phi_r$

$\Phi^\prime$ is polynomial in the size of $\Phi$, and has inputs  $\Pi_{q_{r,1}}$ \ldots $\Pi_{q_{r,Q}}$ as well as all the auxilliary bits. $\Phi^\prime$ also has some special properties:
\begin{itemize}
\item $\Phi$ $\in$ 3SAT $\Longrightarrow$ $\Phi^\prime$ $\in$ 3SAT, since V will accept $\Phi$ for any r $\in$ $\{0, 1\}^R$
\item $\Phi$ $\notin$ 3SAT $\Longrightarrow$ $\underset{r \in \{0,1\}^R}{Pr}$[V rejects] $>$ $\frac{1}{2}$ \newline
\indent $\Longrightarrow$ $\forall$ $\Pi$, at most $\frac{1}{2}$ of the $\phi_r$ clauses are satisfiable (from aux bits) \newline
\indent $\Longrightarrow$ $\forall$ $\Pi$, $\forall$ $aux$, there is at least one unsatisfied clause in at least $\frac{1}{2}$ of the $\phi_r$ clauses \newline
\indent $\Longrightarrow$ at least a $\frac{1}{2S}$ fraction of the clauses of $\Phi^\prime$ are unsatisfiable \newline
\indent $\Longrightarrow$ the maximum fraction of satisfiable clauses of $\Phi^\prime$ is (1 - $\frac{1}{2S}$)
\indent $\Longrightarrow$ approximating 3SAT within a factor of $\frac{1}{4S}$ is NP-hard, where S is some constant dependant on Q
\end{itemize}

\end{document}
