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\begin{document}
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\lecture{14}{Apr 27, 2009}{Jayalal Sarma M.N.}{Jing He}
In previous lectures we saw different approaches for proving
$\textbf{PARITY} \not\in \mathbf{AC^0}$. Today we will introduce
another type of circuit lower bounds, namely, lower bound for monotone
circuits. In 1985, Razborov proved that the \textbf{CLIQUE} problem
does not have polynomial-sized monotone circuits. Before this, we
familiarize ourselves with some ideas related to montonicity itself.
We need some definitions and notations first.
\section{Monotone Circuits}
\begin{definition}[Monotone circuits]
\label{def_mon} A Boolean circuit (over $\{\land, \lor, \lnot\}$)
is called \emph{monotone} if it does not contain any \textbf{NOT} gates.
\end{definition}
\begin{definition}[Monotone Functions]
\label{def_mon_fun}
A function $f:\{0,1\}^n \rightarrow \{0,1\}$ is monotone if and only
if $(\forall x \leq y) f(x) \leq f(y)$, where the "$\leq$" is
performed bitwisely.
\end{definition}
\begin{proposition}
A function $f:\{0,1\}^n \to \{0,1\}$ is \emph{monotone} if and only if
it can be computed by a monotone circuit.
\end{proposition}
\begin{proof}
Monotone functions are closed under composition. That is if for $f$
and $g$ are monotone, then $f(g(u_1),g(u_2), \ldots g(u_k)$ is montone
too. Since $\land$ and $\lor$ are montone it follows that monotone
circuits can compute only monotone functions.
To see the other direction : first we define a (partially) monotone
circuit for the comparator function. That is given, $x,\alpha \in
\{0,1\}^n$, the circuit checks if $x \ge \alpha$. If we fix $\alpha$,
this function is monotone in $x$. It is easy to construct a monotone
circuit for it too.
Now let $f$ be a montone function. Consider the Boolean Lattice with
$1^n$ as the maximum element and $0^n$ as the minimum. Any path from
$0^n$ to $1^n$ has a point where the function $f$ turns from 0 to
1. There are only $p=2^n$ vertex disjoint paths, and this defines a
boundary $\alpha_1 \ldots \alpha_{p}$ between 1-region and
0-region. To decide the function, the circuit has to essentially check
if the given input $x$ is greater than any of these $\alpha_i$'s. Now
the montone circuit will have an $\lor$ gate on top with exponential
fan-in, followed by comparator circuits. This gives the proof. A point
to note is that the size of the circuit that we described is not
polynomial in the input.
\end{proof}
Now we formalize the problem we will address in this lecture. Remember
that an undirected graph $G$ with $n$ vertices can be encoded with a
binary string of length $n\choose{2}$, each bit of which indicates
whether the corresponding edge exists. We use this encoding to define
the problem $\mathbf{CLIQUE_{k,n}}$. Let $G(V,E)$ be a graph on $n$
vertices. Clearly, $G$ can be represented by a bit string
$x_1,x_2,\ldots,x_{n\choose{2}}$ where $x_i$ is 1 if the $i^{\rm th}$
(of the ${n\choose{2}}$ possible edges).
\begin{definition}
\[ \mathbf{CLIQUE_{k,n}}=\left\{x=(x_1,x_2,\ldots,x_{n\choose{2}})~\left|\right.~
\exists \textrm{ a clique of size $k$ in the graph defined by}~x\right\} \]
\end{definition}
A first observation is that this function is monotone. Indeed, if we
add an additional edge to a graph which already has a clique of size
$k$, that clique does not disappear !. Now, by the above argument
about monotone circuits for monotone functions, there is monotone
circuit of size $2^{{n\choose{2}}}$ computing this function. Indeed, a
very similar arguement gives slightly better upper bound.
\begin{proposition}
$\mathbf{CLIQUE_{k,n}}$ can be computed by a monotone circuit of size $O(n^k)$.
\end{proposition}
\begin{proof}
Trivial, for all subsets of size $k$ (there are ${n \choose k}=O(n^k)$
of them), and for each subset cheque whether the edge is present or
not. This can be done by a monotone circuit. The size of the entire
circuit is still $O(n^k)$. Notice that when $k = O(n)$ this gives an
$n^n$ bound which is not polynomial sized.
\end{proof}
Now we will consider lower bounds for the problem
$\mathbf{CLIQUE_{k,n}}$. We will see that the above bound is tight
upto a $\sqrt{k}$ factor in the exponent.
\section{Lower Bounds for CLIQUE}
We will show the following theorem due to Razborov (1985).
\begin{theorem}\label{lower_b}
Any monotone circuit computing $\mathbf{CLIQUE_{k,n}}$ must have
size $n^{\Omega{(\sqrt{k})}}$.
\end{theorem}
Before proving the above theorem we first give an outline. First,
we want to transform any circuit $C$ computing
$\mathbf{CLIQUE_{k,n}}$ into $C'$ which makes a lot of errors when
computing the same problem. Then we prove that the error made by
any single gate of $C'$ is kind of "small". So by a union bound,
we can get a lower bound for the size of $C'$, which also gives a
lower bound for the size of $C$ if they differ not too much.
To put our idea explicitly we need some notations.
\begin{definition}
~An encoded graph is called a \emph{positive input} of
$\mathbf{CLIQUE_{k,n}}$ if it is a minimal graph containing a clique
of size $k$. Let $\mathbf{PI_{n,k}}$ denote the collection of all
such graphs.
\end{definition}
\begin{definition}
~An encoded graph is called a \emph{negative input} of
$\mathbf{CLIQUE_{k,n}}$ if it is a maximal graph which does not
contain a clique of size $k$. Let $\mathbf{NI_{n,k}}$ denote the
collection of all such graphs.
\end{definition}
It is clear that $|\mathbf{PI_{n,k}}|={n\choose k}$ and
$|\mathbf{NI_{n,k}}|=(k-1)^n$.
\begin{definition}
~A \emph{clique indicator} $I_X$ is a boolean function on graphs of
$n$ vertices which outputs 1 if and only if the induced subgraph of
the input graph on vertex set $X$ is a clique. A
\emph{$(m,l)$-approximator} is a boolean function of form
$\bigvee_{i=1}^{r}I_{x_i}$ where $|X_i|\leq l$ and $r\leq m$.
\end{definition}
Suppose we have a monotone circuit $C$ computing
$\mathbf{CLIQUE_{k,n}}$. We want to transform it into a
$(m,l)$-approximator $C'$ for some fixed $m$ and $l$. We do the
transformation inductively. For a single variable $x_{i,j}$, we
change it into a clique indicator $I_{\{i,j\}}$. For a formula
$F_1 \lor F_2$, suppose $A=\bigvee_{i=1}^{r}I_{X_i}$ and
$B=\bigvee_{j=1}^{s}I_{y_j}$ are the corresponding
$(m,l)$-approximators of $F_1$ and $F_2$, respectively. We know
that $T=A\lor B=(\bigvee_{i=1}^{r}I_{X_i}) \lor
(\bigvee_{j=1}^{s}I_{y_j})$, but this is a $(2m,l)$-approximator.
To compress it we need the following lemma from Erdos and Rado:
First, some terminology. A \emph{sunflower} is a collection of $p$
sets $\{Z_1,\ldots,Z_p\}$ such that $\forall 1\leq i\leq j\leq p, Z_i
\cap Z_j = Z$, where $Z$ is called the \emph{center} of the
sunflower. We also call it a $p$-petal sunflower. The choice of these
names are more-or-less self explanatory.
\begin{lemma}[Sunflower Lemma]
\label{sunf}
Suppose $S=\{S_1,\ldots,S_k\}$ is a collection of sets for which
$(\forall 1\leq i\leq k)|S_i|\leq l$ and $k\geq (p-1)^l\cdot l!$, then
there exists a $p$-petal sunflower in $S$.
\end{lemma}
We will include a proof of the Lemma later in this draft. First we
see the application. Now we choose $m=(p-1)^l \cdot l!$, where the
values of $p$ and $l$ will be decided later. If $r+s