\documentclass{article}
\usepackage{amssymb}

\begin{document}
\input{preamble.tex}
\lecture{21}{April 25, 2005}{Madhu Sudan}{Paul Valiant}

\newcommand{\UNSAT}{\rm UNSAT }


\section{Outline}

Today we finish the overview of Dinur's combinatorial proof of the
PCP theorem.

Recall the problem max-$k$-SAT-$\Sigma$, an instance of which takes
the following form.

\begin{itemize}
\item{We have $n$ variables $x_1,...,x_n$, taking values in $\Sigma$.}
\item{We have $m$ constraints $C_1,...,C_m$, each constraint $C_i$
acting on at most $k$ variables via an arbitrary predicate
$P:\Sigma^k\rightarrow \{0,1\}$.}
\end{itemize}

The task is to find an assignment to the variables that maximizes
the number of satisfied constraints.

Recall from last lecture, that the PCP theorem is equivalent to the
statement that one can transform SAT problems $\phi$ into
max-$k$-SAT-$\Sigma$ problems $\phi'$ that are robust in the
following sense:
\begin{quote}
There exists a constant $\epsilon$ such that
  \begin{itemize}
  \item{If $\phi$ is satisfiable, $\phi'$ is (perfectly) satisfiable.}
  \item{If $\phi$ is not satisfiable, then at least $\epsilon$ fraction
  of the clauses of $\phi'$ must be unsatisfied for any assignment to
  the variables of $\phi'$.}
  \end{itemize}
\end{quote}

For a problem $\phi$, denote by $\UNSAT(\phi)$ the minimum nonzero
fraction of clauses of $\phi$ that may be unsatisfied. The idea is
that if we can transform any 3SAT problem into a problem $\phi'$
with $\UNSAT(\phi')\geq \epsilon$, then we can
\emph{probabilistically} test a claimed assignment to $\phi'$ by
testing a random clause; this test would have soundness
$1-\epsilon$.  Thus since 3SAT is NP-complete, we can construct such
PCPs for any problem in NP.

We note that for a SAT problem $\phi$ with $m$ clauses, the
unsatisfiability gap $\UNSAT(\phi)$ is at least $\frac{1}{m}$. The
main idea of Dinur's proof is a technique to \emph{amplify} this
fraction to a fixed constant $\epsilon$.  This takes the form of the
following lemma.

\begin{lemma}[main]
There exists a transformation $T$ mapping max-2-SAT-$\Sigma$ into
itself with the following properties:
\begin{itemize}
\item{$T$ increases the number of variables and clauses by an at
most linear factor.}
\item{$\UNSAT(T(\phi))\geq\min\{\epsilon,2\UNSAT(\phi)\}$.}
\end{itemize}
\end{lemma}

Iteratively applying this lemma $\log m$ times will increase
$\UNSAT(\phi)$ to a constant $\epsilon$, while increasing the size
of $\phi$ by at most polynomial factor.

The proof of this lemma is divided into two sub-lemmas.  See the
previous lecture notes for details.

\begin{lemma}[Lemma 1]\label{lemma 1}
There is a reduction $T_1$ from max-2-SAT-$\Sigma$ to
max-2-SAT-$\Sigma^l$ for some $l$ such that the unsatisfiability gap
becomes ``much'' larger.
\end{lemma}

\begin{lemma}[Lemma 2]\label{lemma 2}
There is a reduction $T_2$ from max-2-SAT-$\Gamma$ into
max-2-SAT-$\Sigma$, that possibly reduces the unsatisfiability gap,
but by a fixed constant that is independent of $\Gamma$.
\end{lemma}

In turn, Lemma \ref{lemma 2} follows from the following two smaller
lemmas.

\begin{lemma}[Lemma 2a]\label{lemma 2a}
There is a reduction $T_{2a}$ from max-2-sat-$\Gamma$ to
max-$k$-SAT-$\{0,1\}$, for some $k$. \end{lemma}
\begin{lemma}[Lemma 2b]\label{lemma 2b}
There is a reduction $T_{2b}$ from max-$k$-SAT-$\{0,1\}$ to
max-2-SAT-$\Sigma$.
\end{lemma}

Lemma \ref{lemma 2b} was shown last time.  Here we outline the
proofs of Lemmas \ref{lemma 1} and \ref{lemma 2a}.

\section{Gadgets}
The proofs of these lemmas are going to take the form of
``gadgets''.   The general form of such a gadget (for our purposes)
is the following.

We are given a constraint satisfaction problem $\phi$ with variables
$\{x_i\}$ and clauses $\{C_j\}$, and wish to transform the problem
so as to meet some specified criteria.

We do this via an encoding function $E$, that transforms $\phi$ to
$\phi'$ as follows:
\begin{itemize}
\item{For each variable $x_i$, $E$ maps $x_i$ into a set of
variables $x'_{i,1},...,x'_{i,t}$, for some $t$.}
\item{For each constraint $C_j$, $E$ introduces auxiliary variables
$y_{j,1},...,y_{j,u}$ for some $u$.}
\item{Then for $E$ maps each constraint $C_j$ into a set of new constraint
$C'_{j,1},...,C'_{j,v}$ on some subset of the new variables
$\{x',y\}$}.
\end{itemize}
The encoding function $E$ must preserve satisfiability: if $\phi$ is
satisfiable, then so is $\phi'$.  Furthermore, to relate the
satisfaction of clauses of $\phi'$ back to those of the original
problem, we introduce a \emph{decoding} function $D$, that will
\emph{decode} variable assignments of $\phi'$ to variable
assignments of $\phi$ in such a way that if a large fraction of the
clauses $C'_{j,1},...,C'_{j,v}$ are satisfied, then the $C_j$ will
be satisfied on the decoded assignment $D(x'_{i,1},...)$.

Such gadgets are fundamental to the following proofs.

\section{Proof of Lemma 2a}
Recall that we are trying to reduce max-2-SAT-$\Gamma$ to
max-$k$-SAT-$\{0,1\}$.  When constructing an encoding $E$, as
described above, we clearly need to represent a variable $x_i$ in
$\Gamma$ by at least $\log|\Gamma|$ binary variables $x'_{i,k}$.  We
however choose to use many more than $\log|\Gamma|$ variables.

Recall that in the reduction we are constructing, we aim to preserve
(up to linear factors) the unsatisfiability gap.  In other words, if
a high percentage of the new constraints $C'$ are satisfied, then we
want a guarantee that a comparably high percentage of the original
constraints $C_j$ are satisfied.

We are motivated by the PCP construction to express the
probabilistically checkable constraints in terms of tables of
function evaluations.  The gadget is constructed as follows.

For each variable $x_i$, construct variables $x'_{i,f}$ for
\emph{every} function $f:\Gamma\rightarrow\{0,1\}$.  We will later
construct constraints that will let an assignment $\{x'_{i,f}\}$ be
interpreted as the result of $f$ being applied to the
\emph{original} variable $x_i$.

Recall that each constraint $C_j$ consists of a predicate
$P(x_{i_1},x_{i_2})$ operating on two of the (original) variables.
 For each such constraint $C_j$, we construct auxiliary variables $y_{j,g}$
for every function $g:\Gamma\times\Gamma\rightarrow\{0,1\}$.  As
above, we will construct constraints so that if a high fraction of
the constraints on the transformed variables $x'_*,y_*$ are
satisfied then the values $y_{j,g}$ may be correctly interpreted as
the values of $g$ on a pair $(a,b)$, which will be seen to satisfy
the $j$th constraint of the original problem.

The constraint $C_j$ is transformed into a set of constraints
$C'_{j,\{g_1,g_2,g_3,f\}}$ for \emph{every} quadruple of functions
$\{g_1,g_2,g_3,f\}$.  The constraint $C'_{j,\{g_1,g_2,g_3,f\}}$ is
the conjunction of five constraints, defined below.

The first two constraints ensure that the variables $y_{j,*}$ define
proper functions:
\begin{itemize}
  \item[1.] {$y_{j,g_1}+y_{j,g_2}=y_{j,g_1+g_2}$}
  \item[2.] {$y_{j,g_1}\cdot y_{j,g_2}=y_{j,g_1g_2+g_3}-y_{j,g_3}$}
\end{itemize}
The first test evidently ensures that function evaluation commutes
with addition.  The second test does the same for multiplication,
with the slight caveat of adding and subtracting the function $g_3$.
Note that if the test omitted $g_3$ then both sides of the
constraint would be 0 with probability $\frac{3}{4}$, which would
make the distribution of the queries $y_{j,g_1g_2}$ significantly
non-uniform, and would let us ``cheat'' when assigning variables to
$y_{j,*}$.

The third test will ensure that the pair $(a,b)$ represented by
$y_{j,*}$ actually satisfies $P$.  One deterministic way to do this
is to note that $P$ is a function from $\Gamma\times\Gamma$ into
$\{0,1\}$, so we can evaluate $y_{j,P}$ and constrain it to be 1.
But as above, this test would not be robust enough since it samples
$y_{j,*}$ in a highly non-uniform manner.  We modify it to the
following:
\begin{itemize}
  \item[3.] {$y_{j,P+g_3}-y_{j,g_3}=1$}
\end{itemize}

We now note that even if we have succeeded so far in transforming
each constraint $C_j=P(x_{j_1},x_{j_2})$ into a way to express the
variables $x_{j_1},x_{j_2}$, we must still ensure that when the same
variable $x_i$ appears in multiple constraints these representations
correspond to a consistent value in $\Gamma$.  We do this by
relating the variables $y_{j,*}$ to the variables $x'_{j_1,*}$ and
$x'_{j_2,*}$.

Given a univariate function $f$, let $f^{(1)}$ denote the bivariate
function $f^{(1)}(a,b)\eqdef f(a)$ that uses only its first input.
Define $f^{(2)}$ similarly to be the function that uses only its
second input.  We create the following two constraints to ensure
consistency of all the representations of the original variables:
\begin{itemize}
  \item[4.] {$y_{j,f^{(1)}+g_3}-y_{j,g_3}=x'_{j_1,f}$}
  \item[5.] {$y_{j,f^{(2)}+g_3}-y_{j,g_3}=x'_{j_2,f}$}
\end{itemize}
From here, Dinur shows that if at least some fixed fraction of the
constraints $C_{j,*}$ are satisfied then the involved variables can
be ``decoded'' to a satisfying assignment of the original problem.
If this fraction were (say) 99\%, then this would imply that the
unsatisfiability gap decreases by at most a factor of 100 under this
transformation.  We omit the details and move on to a proof of Lemma
1.

\section{Proof of Lemma 1}
We show how to amplify the unsatisfiability gap of a
max-2-SAT-$\Sigma$ problem. We motivate our approach with some
natural ideas that do not quite work.

The most natural approach is to replace constraints of the original
problem with \emph{pairs} of constraints.  Thus for every pair
$C_{j_1},C_{j_2}$ we would form a new constraint that consists of
the conjunction $C_{j_1}\wedge C_{j_2}$.  If at most $1-\epsilon$
fraction of clauses were satisfied in the original problem, then at
most $(1-\epsilon)^2$ fraction will be satisfied in the new problem,
which will essentially double the unsatisfiability gap.

However, note that after the above transformation, constraints now
operate on \emph{four} variables at a time, instead of the required
two.  A second idea is to now replace variables with pairs of
variables, so that each constraint will now operate on a pair of
pairs of variables, instead four individual variables.  In this
manner, we could transform max-2-SAT-$\Sigma$ problems into
max-2-SAT-$\Sigma^2$ problems, but we might no longer be sure about
the unsatisfiability gap.  It turns out there is a theorem called
the ``Parallel Repetition Theorem'' that would imply the desired
result if instead of taking pairs of everything, we took $l$-tuples,
for large enough $l$.  This approach, however, introduces nonlinear
blowup to the number of constraints and variables, which we cannot
afford.

The problem here is that we are trying to sample random $l$-tuples
by using $l$ times as many (random) index bits, and we cannot afford
linear blowup in the number of index bits.  The idea is to use some
kind of pseudo-random generator to get the effect of $l$-tuples
using a generator that uses only \emph{additively} more bits.

\subsection{Amplification of BPP}
In this section we recall a standard result on how to amplify the
success probability of BPP algorithms using ``recycled'' randomness.

Let $L$ be some language in BPP.  By definition, there exists a
Turing machine $A$ using $r(n)=\poly(n)$ random bits such that
\[\forall x\in \{0,1\}^n, \Pr_{R\leftarrow
\{0,1\}^{r(n)}}[A(x,R)=L(x)]\geq\frac{2}{3}.\]

We note that using more random bits, we can amplify this
probability.  Specifically, using $q(n)r(n)$ random bits, for some
polynomial $q$, we can increase the success rate from $\frac{2}{3}$
to $1-2^{-q(n)}$ by repeating the above experiment $q$ times and
taking the majority outcome.

The drawback here, is that we require $q$ times more randomness.

A natural idea to try is to generate \emph{pairwise} random numbers.
We can generate pairwise independent numbers from a seed consisting
of just a pair of random numbers, and thus use only $2r(n)$ random
bits.  However, because the resulting pseudo-randomly generated
numbers are not independent, the success rate does not amplify as
above.  Specifically, if for fixed $x$ we consider $\{0,1\}^{r(n)}$
as being divided into two sets, on one of which $A$ returns the
right answer, on one of which $A$ returns the wrong answer, we want
a guarantee that not too many of the pseudo-random samples will lie
in the ``wrong'' set. We would like to apply something like the
Chernoff bound, but unfortunately that assumes complete
independence.

\subsection{Expander Walks}
A more hopeful idea is to construct a sequence of numbers in
$\{0,1\}^{r(n)}$ by taking a random walk on a graph with vertex set
$\{0,1\}^{r(n)}$.  If the degree of our graph is small, some
constant $d$, then each new sample requires only $\log d$ more bits
to specify the next leg of a random walk.

Explicitly, given a graph $G$, a random walk on $G$ is a sequence
$R_j$ constructed as follows.

\begin{itemize}
  \item Pick $R\in G$ at random.
  \item Pick $q(n)$ choices $i_1,...,i_{q(n)}\in \{1,...,d\}$ at
  random.
  \item Let $R_1=R$, and iteratively define $R_j$ to be the $i_j$th
  vertex adjacent to $R_{j-1}$.
\end{itemize}

We ask here whether such a random walk will necessarily produce
elements of the right independence properties.  We immediately note
that if the graph $G$ is composed of two disjoint pieces, then the
choice of the starting point will determine which half the rest of
the walk falls into, and that such such walks will have very bad
properties for us. We want to ensure that such a case does not
happen, or in other words, that the graph is very well connected.

It turns out that expander graphs provide the desired properties.

Recall that a graph $G=(V,E)$ is called an $\epsilon$-expander if
for all sets of vertices $S\subset V$ that are sufficiently small,
$|S|\leq \frac{|V|}{2}$, the number of edges crossing from $S$ to
its complement is large: \[|E(S,\overline{S})|\geq\epsilon|S|.\]

We claim the following:

\begin{claim} If $G$ is an $\epsilon$-expander then a random walk on
$G$ hits every set $S$ roughly $\frac{|S|}{|V|}$ fraction of the
time, where ``roughly'' is interpreted to correspond to some kind of
Chernoff bound.
\end{claim}

This means that with $r(n)+d\cdot q(n))$ randomness, we can increase
our probability of success in an exponential manner by applying
these Chernoff-type bounds.

\section{Dinur's Amplification}
We now briefly outline how this technique may be applied to the
max-SAT problem.

Given a max-2-SAT problem $\phi$ with variables $x_1,...,x_n$ and
constraints $C_1,...,C_m$, define the \emph{constraint graph} $G$ of
$\phi$ to be the graph on $n$ vertices and $m$ edges such that each
vertex corresponds to a variable $x_i$, and each constraint becomes
an edge that connects the two variables involved in the constraint.

We then ``convert'' $G$ into a $d$-regular $\epsilon$-expander
graph.  The construction proceeds in two steps: the first step is to
reduce the degree of $G$ to a constant $d$ without changing $G$
``too much'' (this is a zig-zag product); the second step is to add
edges until the graph has good expansion properties.  Denote this
version of $G$ by $G'$.

Our plan is now to take a random walk on $G'$, and verify the
constraints corresponding to every edge we cross.  Because of the
above claim about expanders, such a choice of constraints will be
suitably random.

Recall however, that since many vertices of $G'$ may correspond to
the same vertex of $G$, we have to add a compatibility check to
ensure that a variable $x_i$ takes consistent values in its
different incarnations.

To do this, we apply the familiar trick of replacing $G'$ with some
\emph{much} huger graph $G''$ that contains enough redundancy to
allow for probabilistic verification.

Specifically if $t$ is the length of our intended random walk, let
$G''$ be the multi-graph whose vertices are the same as those in
$G'$, but which has an edge for every \emph{path} of length at most
$t/2$ in $G'$.  Furthermore, we append to the label of each vertex
of $G''$ the label in $G'$ of each of its $\leq d^{t/2}$ neighbors.
We note that $G''$ has alphabet $\Sigma^{d^{t/2}}$.

The constraint associated with an edge $(u,v)$ of $G''$ is satisfied
if all of the assignments in $G'$ at the intersection of the radius
$t/2$ neighborhoods of $u$ and $v$ are consistent, and satisfy all
their constraints from $G'$.

Note that while such checks involve a huge amount of data, all this
data is contained in the labels of the vertices $u,v\in G''$, so
this is still an instance of max-2-SAT.

We omit the various robustness proofs for this procedure, but note
the following ``decoding'' algorithm that will help show that
anything almost-satisfying $G''$ can be decoded to an assignment
that almost-satisfies $G$.

\begin{quote}{\bf Decoding assignments to $G''$:}
Given an assignment to vertices of $G''$, to find a corresponding
assignment to a vertex $u\in G'$, take a random walk of length $\leq
\frac{t}{2}$ from $u$, and whichever vertex $v$ we end up at, find
its corresponding vertex in $G''$, and inspect its label and return
its opinion of the value of $u$.
\end{quote}

In the next lecture we will consider ``recycled randomness'' in more
depth.


\end{document}
