\documentclass[10pt]{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[pdftex]{graphicx}

\setlength{\parskip}{10pt}
\setlength{\parindent}{0pt}

\oddsidemargin=0.15in
\evensidemargin=0.15in
\topmargin=-.5in
\textheight=9in
\textwidth=6.25in

\begin{document}
\input{preamble.tex}

\lecture{2}{February 13, 2012}{Ronitt Rubinfeld}{Hamsa Sridhar}

\section{Review}

Last class, we introduced the problem of hypergraph coloring. We found that we could guarantee the existence of a non-monochromatic 2-coloring for the $l$-element sets $S_1,...S_m \subseteq S$ using the probabilistic method, given a bound on the number of sets $m < 2^{l-1}$. However, we also noted that we could likely find a weaker constraint if all the sets $S_1,...,S_m$ were somewhat `independent,' i.e. did not have too many points in common.

{\bf Definition}: An event $A$ is {\bf independent} of $B_1,...B_k$ if $\forall J \subseteq [K]$ such that $J \neq \emptyset$,
\begin{align*}
\text{Pr}[A \cap \bigcap_{j \in J} B_j] = \text{Pr}[A] \text{Pr}[\bigcap_{j \in J} B_j].
\end{align*}

With this notion of independence in mind, we could then define a dependency digraph:

{\bf Definition}: Given the events $A_1,...A_n$, $D=(V,E)$ with $V=[n]$ is a {\bf dependency digraph} of $A_1,...A_n$ if each $A_i$ is independent of the set of ${A_j}$ that don't neighbor it in $D$.

We now introduce the symmetric version of the {\bf Lov\'asz Local Lemma}, which states that:

{\bf Theorem}: Given events $A_1,...A_n$ such that $\text{Pr}[A_i] \leq p$ $\forall i \in [n]$ with dependency digraph $D$ such that $D$ is of degree $\leq d$, if $ep(d+1) \leq 1$, then $\text{Pr}[\bigcap_{i \in [n]} \bar{A_i}] > 0$.

A key point to note is that the Lov\'asz Local Lemma only restricts the degree of $D$ and not the number of vertices $V$. In other words, we are placing a lower bound on the independence of the `bad events' $A_i$, rather than placing an upper bound on the number of $A_i$. Today, we will see this new restriction come into play for the hypergraph coloring problem. We will also begin constructing the Moser-Tardos algorithm for finding a solution to the symmetric case of the hypergraph coloring problem.

\section{Hypergraph Coloring (revisited)}

{\bf Theorem}: Given $l$-element sets $S_1,...S_m \subseteq S$ such that each $S_i$ intersects $\leq d$ other $S_j$'s, if $\frac{e(d+1)}{2^{l-1}} \leq 1$ then there exists a 2-coloring of $S$ such that no $S_i$ is monochromatic.

{\bf Proof}: Color each element of $S$ red or blue independently with probability $1/2$. Define $A_i$ as the event that $S_i$ is monochromatic. Then, the $\text{Pr}[A_i] = 2 \frac{1}{2^l} = \frac{1}{2^{l-1}}$ $\forall i \in [m]$. Moreover, since each $S_i$ intersects at most $d$ other $S_j$'s, $A_i$ depends on at most $d$ other $A_j$'s and its dependency digraph $D$ has degree $\leq d$. Now, we can directly apply the Lov\'asz Local Lemma to conclude that there exists a non-monochromatic 2-coloring as desired.\hspace{\stretch{1}}$\Box$

Once again, we can compare the probabilistic method with the Lov\'asz Local Lemma. Given a hypergraph of $m$ edges with $l$ vertices each, the probabilistic method guarantees a non-monochromatic coloring if we bound the number of edges, i.e. $m < 2^{l-1}$, and the Lov\'asz 
Local Lemma guarantees a non-monochromatic coloring if we bound the degree, i.e. $d+1 \leq 2^{l-1}/e$.

The Lov\'asz Local Lemma is particularly powerful because it guarantees a solution for some sub-classes of NP-hard problems. For instance, we can prove that every Boolean formula $\phi$ in $k$-conjunctive normal form ($k$-CNF) is satisfiable if it has the property that every variable occurs in at most $2^{k-logk-log2}$ clauses of $\phi$. (Recall that $\phi$ is a $k$-CNF if $\phi$ is a conjunction of clauses, where each clause contains at most $k$ literals.)

\subsection{Brief History}

1975: Lov\'asz Local Lemma proposed; non-constructive; bound $d \leq 2^{l-2}$\\
1991: Beck proposed constructive algorithm; bound $d \leq 2^{l/48}$\\
1991-2009: Many results\\
2009: Moser's constructive algorithm for the symmetric case; bound $d \leq 2^{l-O(1)}$\\

Later, Moser and Tardos extended Moser's constructive algorithm to most problems (non-symmetric cases).

\section{Moser-Tardos Symmetric Algorithm}

{\bf Algorithm}: Start with a random coloring of $S$. While there is a monochromatic set, pick an arbitrary `violated' $S_i$ and randomly reassign colors to the elements of $S_i$.

Now, to see why this algorithm works, we define the following:

{\bf Definition}: The {\bf log of execution} is a set of pairs $(1,S_{i1}),(2,S_{i2}),...$ where the first entry of each pair is the `loop number' and the second entry is the name of the the resampled set.

{\bf Definition}: The {\bf witness tree for step $j$} ($j \ge 0$) is constructed as follows -
\begin{enumerate}
\item The root vertex is labeled by $S_{ij}$.
\item Go backwards through the log for $t=j$ down to $t=1$. If $S_{it}$ intersects with any $S_{ij}$ in the witness tree, add $S_{it}$ to the witness tree by making it point to a node $S_{ij}$ with which it intersects that is of maximum depth (picking arbitrarily if there is more than one choice at the maximum depth).
\end{enumerate}

See the example from the class handout (next page). The corresponding log of execution and witness trees are given below:

\begin{center}
\includegraphics[scale=0.5]{witnesstrees.png}
\end{center}

Note that the number of re-samplings is given by the number of trees. Next class, we will bound the number of possible trees and the probability of getting a large tree using indicator variables. This will show that the probability of not getting a non-monochromatic 2-coloring after a large number of re-samplings is extremely small, giving us the desired result.

\includegraphics[scale=0.7]{scan.pdf}

\end{document}
