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%%%  EDIT THE FOLLOWING TO UPDATE DATE AND LECTURE NUMBER

\head{CS G399}{Algorithmic Power Tools I}{Fall 2007}{Scribe: Jingjing Duan}
{9 November 2007}

\newcommand{\p}{\mathcal{P}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\coll}{\mathcal{S}}
\newcommand{\xpy}{x + \lambda y}
\newcommand{\Alg}{\mathcal{A}}
\newcommand{\opt}{\mbox{OPT}}
\newcommand{\eps}{\varepsilon}

\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

\noindent {\bf Lecture Outline:}
%%% Itemized outline of the lecture
\bi
\item SDP
\item MAX-CUT
\ei

Last class, we discussed fractional chromatic and clique number. We showed that
\[ \omega(G) \le \mbox{sup} \sqrt[n]{\omega(G^n)} = \mbox{L}\Theta(G)
\le \chi_f(G) \le \chi, \] where $\mbox{L}\Theta(G)$, the Lovasz theta
function of $G$, is the largest value of $\sqrt{\frac{1}{\cos
\theta}}$ achieved by a nice orthonormal representation of $G$ (recall
that all the vectors in a nice orthonormal representation subtend an
angle $\theta$ with a particular unit vector $e$).

We now consider an alternative defintion of the fractional chromatic number.

\begin{Definition}
  An $(a,b)$-fold coloring is an assignment $S: V \rightarrow 2^U$,
  $|U| = b$, such that $|S(v)| = a$, for all $v$, and $S(v_1) \cap
  S(v_2) \neq \emptyset$ if $(v_1, v_2) \in E$.  The $a$-fold
  chromatic number equals $\min \frac{b}{a}$, where the minimum is
  over all $(a,b)$-fold colorings.
\end{Definition}

An example makes this clear.
\begin{figure}[htb!]
  \begin{center}
  \includegraphics[height=60mm]{pentagon.eps}
  \end{center}
\end{figure}

In other words, assign objects to each node such that two connected nodes
don't share any objects.

Clearly, an optimal $(1,b)$-fold coloring gives the normal chromatic
number.

\begin{Definition}
A graph $G$ is $(a,b)$-choosable if for any assignment of a list of
$a$ objects to each of its vertices there is a subset of $b$ objects
of each list so that subsets corresponding to adjacent vertices are
disjoint.  The $a$-choosable chromatic number of $G$ is $\min
\frac{b}{a}$, over all $(a,b)$ such that $G$ is $(a,b)$-choosable.
\end{Definition}

It can be shown that the fractional chromatic number $\chi_f$, the
best $a$-fold chromatic number, and the best $a$-choosable chromatic
number are all equivalent.  Formally,
\[
\inf_a \mbox{$a$-fold coloring} = \inf_a \mbox{$a$-choosable chromatic
number} = \chi_f.
\]

\section{SDP}
Semidefinite programming (SDP) is a subfield of convex optimization concerned
with the optimization of a linear objective function over the intersection
of the cone of positive semidefinite matrices with an affine space.

A simple comparison between LP and SDP:
\[ \begin{array}{c|c}
  \mbox{LP} & \mbox{SDP} \\ \hline
  \ge & \succeq \\
  \mbox{ vector } \cdot & \mbox{ matrix } \cdot \\
  \end{array} \]
  
In SDP, we have Frobenius inner product of two matrices, which is the 
component-wise inner product of two matrices as though they are vectors.
In other words, it is the sum of the entries of the Hadamard product, that is,
\[ A \cdot B = \sum_i \sum_j A_{ij}B_{ij} = \mbox{trace}(A^TB) = 
\mbox{trace}(AB^T) \]

Further, the trace of a square matrix $A$ is defined to be the sum of the
elements on the main diagonal of $A$, i.e., 
\[ \mbox{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} = \sum_i a_{ii} \]

Primal-dual of SDP:
\[\begin{array}{c|c}
  \begin{array}{c}
    \min \ C^Tx \\
    \sum x_i A_i \succeq B \\ \\
  \end{array} &
  \begin{array}{c}
    \max \ yB \\
    yA_i = C_i \\
    y \succeq 0 \\
  \end{array}
\end{array} \]

Here $A$ and $B$ are positive semidefinite matrices. They have to be 
symmetric. They have the following properties:
\bi
\item $\forall x, x^T A x \succeq 0$
\item eigen values are non-negative
\item $A = U^TU$, for some $U$
\item $A$ is a non-negative linear combination of $xx^T$, for some vector $x$
\item symmetric minor determinant is non-negative
\ei

A little example:
\[\begin{array}{ccc}
  \begin{array}{c}
    \min x_1 \\
    \left( \begin{array}{cc}
        x_1 & 1 \\
        1 & x_2 \\
        \end{array} \right) \succeq 0 \\
  \end{array}
  & \Longleftrightarrow &
  \begin{array}{c}
    \min x_1 \\
    x_1 x_2 \succeq 1 \\
    x_1, x_2 \succeq 0 \\
  \end{array}
\end{array} \]

SDP is equivalent to Vector Programming. A vector program is a program in which
we optimize functions of linear inner products of vectors subject to linear
constraints. More specifically, a Vector Program can be formulated as follows.
\[ \min \sum_{1 \le i,j \le n} c_{ij}v_i^Tv_j \]
\[ \sum_{1 \le i,j \le n} a_{ij}^{(k)}v_i^Tv_j = b_k, 1 \le k \le m \]
\[ v_i \in \mathcal{R}^n, 1 \le i \le n \]

\section{MAX-CUT}
The Max-Cut problem is defined as: given a graph $G=(V,E)$, a weight function
$w: E \to Z^+$. The goal is to determine a cut $(S,\overline{S})$, where $S \ne 
\emptyset \mbox{ or } V$, such that $w(S, \overline{S})$ is maximized.

A straightforward greedy algorithm will give a $\frac{1}{2}-$ratio
approximation algorithm: \bi
\item Suppose $V$ will be divided into 2 subsets $V_1$ and $V_2$. Initially,
  $V_1$ is empty.
\item Pick the node of the highest degree and put it into $V_1$.
\item See whether it increases the weight of edges crossing.
\item Keep doing this.
\ei

There are 2 other ways to approximate the max-Cut problem:
\bi
\item Local improvement: Start from an arbitrary partition and
repeatedly move a vertex from one side to the other if the max-cut improves.
\item Random Optimization: Randomly partition the nodes into two sets.
\ei
They also give a $\frac{1}{2}$-ratio approximation.

An attempt at writing an LP for Max-Cut problem would be:
\[ \max_{e=ij} \sum y_{ij}\]
such that
\[ y_{ij} \le |x_i - x_j|\]
\[ 0 \le x_i \le 1 \]
Actually, it doesn't quite work out, since it is not easy to deal with 
$|x_i - x_j|$.

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