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\head{CS G399}{Algorithmic Power Tools I}{Fall 2007}{13 November
2007}{Abhishek Chaubey}


\newcommand{\p}{\mathcal{P}}
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\noindent {\bf Lecture Outline:}
\begin{itemize}
\item
Goemans-Williamson Algorithm for Max-Cut
\end{itemize}

In this lecture we will use semi-definite programming (SDP) approach
as a black box to solve the Max-Cut problem. The SDP machinery used
to solve this problem has developed the whole new class of approach
to solve a bunch of problems. Both the algorithm and analysis by
Goemans and Williamson are very simple and short.

The approach would be to write a Strict Quadratic Program for
Max-Cut and convert it to Vector Program. And then solve the Vector
Program using SDP solver. In fact most of the machinery that has
been developed using SDP in approximation algorithms uses vector
programs.


\section{Introduction}

In this section we introduce and review some notions related to
various mathematical programs.

\begin{itemize}
\item
Semi-Definite Programs where we have positive semi-definite matrix
constraints.

\item
Strict Quadratic Programs are quadratic programs (i.e. the
constraints and the objective function could be quadratic) where sum
of power of the variables in each term is either 0 or 2.

\item
Vector Programs (refer to last lecture)

\end{itemize}

{\bf Relationship of Vector Programs with Strict Quadratic
Programs:} In Vector Programs, in each constraint and objective
function, each term is of the form constant $c$ or $v_i$.$v_j$ where
$v_i$ and $v_j$ are vectors and operation is dot product. We can
take a Strict Quadratic Program, replace the scalars (or variables)
by vectors to get Vector Program.

{\bf Machinery to solve a problem:} Take the problem, write it as
Strict Quadratic Program and then replace the variables by vectors
to get Vectors. And we know the fact that semi-definite programming
can solve Vector Programs.

\section{Max-Cut Problem}

{\bf Problem:} Given a graph $G = (V,E)$ and $w: E \rightarrow Z^+$.
Determine cut $(S,\bar{S})$ where $w(S,\bar{S})$ is maximized.

\subsection{Algorithms}

There can be various approaches to solve Max-Cut Problem:

{\bf (i) Local Greedy Algorithm:}

We discussed a 1/2-approximation algorithm in the last class. In
this approach, we start with a cut and greedily try to improve the
size of the cut.

{\bf (ii) "Naive" Randomized Algorithm:}

Place $v$ in $S$ independently with probability 1/2.
\[
Pr[(u,v) \mbox{ crosses the cut}] = 1/2
\]

By linearity of expectation,
\[
E[w(S,\bar{S})] = \frac{\sum_{i>j}w_{ij}}{2}
\]

\begin{eqnarray}
\mbox{Approximation Ratio} & = & \frac {E[w(S,\bar{S})]}{OPT} \nonumber\\
                           & = & \frac {\displaystyle \frac{\sum_{i>j}w_{ij}}
                                                           {2}}
                                                      {OPT}\nonumber\\
                    & \ge & \frac{\displaystyle \frac{\sum_{i>j}w_{ij}}
                                                     {2}}
                               {\sum_{i>j}w_{ij}}\nonumber\\
                    & = & 1/2
\end{eqnarray}

So here we have the trivial algorithm, in which we are not even
looking at the graph. To get (1), we bounded $OPT$ i.e. $OPT$ is at
most $\sum_{i>j}w_{ij}$. Now, if we can provide a better bound for
$OPT$, we will get a better approximation ratio. In this analysis
the bound for $OPT$ is naive. One attempt to get a good
approximation ratio is to improve this bound. So, there is a
possibility that this algorithm might be a better algorithm, but our
analysis is not perfect. A question to address: Is the algorithm
really an 1/2-approximation? One can easily come up with an example
where this algorithm (or minor variant) is indeed an
1/2-approximation.


{\bf (iii) Goemans Williamsons Method:}

It seems very hard to write an LP for Max-Cut problem. One way to
write a LP can be the following:

We have variable $x_{S,\bar{S}}$ for cut $(S,\bar{S})$.

\[
 \max x_{S,\bar{S}} \cdot w_{S,\bar{S}}
\]
\[
 \mbox{such that } \sum x_{S,\bar{S}}=1
\]

It is not a very useful LP. This LP is just an enumeration of all
the cuts. It doesn't capture any structure of the graph. And also we
don't know how to solve it in polynomial time.

We should move away from restricting our notion to LPs because there
can be more problems which can be written in more general programs
than LPs and can be solved in polynomial time. So we should not
constraint ourself to LPs.

One another approach might be to have a variable $x_e$ for each edge
$e$ such that $x_e$ = 1 if it crosses the cut and 0 otherwise.
Objective function would be to maximize $\sum_{e} w_e.x_e$. What are
our constraints? These edges should define a cut.

Now instead of assigning variables for cuts or edges, we will assign
a variable to each vertex and write a strict quadratic program as
follows:

Assign variables to vertices (say $y_i$ for vertex $i$).

\[
y_i = \left\{ \begin{array}
      {r@{\quad:\quad}l}
+1 & $if $ i \in S \\ -1 & $if $ i \notin S
      \end{array} \right.
\]

Strict Quadratic Program [Q][MC]:

\[
 \max \sum_{i > j} w_{ij} \frac{1-y_i.y_j}{2}
\]
\[
 \mbox{such that } y_i \in \{-1,1\}
\]

Note that if $(i,j)$ $\in$ S, then $\frac{1-y_i.y_j}{2}$ =1. Also,
we can write $y_i$ $\in$ \{-1,1\} as $y_i^2$ = 1. So we have written
a Strict Quadratic Program (SQP) for our problem.

Goemons and Williamson relaxed the notion of scalars in [Q] by
vectors and wrote the following vector program.

Vector Program [P]:

\[
 \max \frac{1}{2} \sum_{i > j} w_{ij} \cdot (1-v_i.v_j)
\]
\[
 \mbox{such that } v_i \cdot v_i = 1
\]

Since there are $n$ vertices, we have $n$ vectors. W.l.o.g we can
can assume these vectors to be $n$-dimensional i.e. $v_i$ $\in$
$R^n$ . Note that the constraint $v_i \cdot v_i$ = 1 says that all
these $n$-dimensional vectors are on an $n$-dimensional sphere. Now
we know how to solve this Vector Program in polynomial time using
SDP.

\[
Z_{MC}^* = Z_Q^* = \mbox{Optimal Max Cut}
\]

$Z_P^*$ can be found by SDP to an additive error of $\epsilon$.


So the vector program [P] is solved by SDP solver and we get the
solution, say $(v_1,v_2,\ldots,v_n)$. Now we need to round this
solution to a cut.

If all these vectors are in 1-dimension, then if the value is +1, we
will put in $S$ and if -1 then we put it in $\bar{S}$. So, in this
case we get cut of value $Z_P^*$. Note that $Z_P^* \ge Z_{MC}^*$,
since any solution to [MC] is a solution to [P]. Note that this is
the place where we are getting a better upper-bound than just taking
the total weight of all the edges (which we did in (1)).

\begin{tabbing}
$\mbox{\sc Algorithm}$\\
1. Solve vector program [P] and get optimal solution $(v_1,v_2,\ldots,v_n)$\\
2. Pick random hyperplane $H$ (or a random vector $r$) \\
3. If $v_i \cdot r > 0$, put vertex $i$ in $S$ else put in
$\bar{S}$.
\end{tabbing}

{\bf Analysis:}


Let $\theta$ be the angle between $v_i$ and $v_j$.

\begin{figure}[htb!]
  \begin{center}
  \includegraphics[height=60mm]{cut.png}
  \end{center}
\end{figure}

\begin{eqnarray}
Pr[(i,j) \mbox{ is in cut}] & = & \frac {\theta}{\pi} \nonumber\\
                            & = & \frac {\cos^{-1}{(v_i \cdot v_j)}}{\pi}
\end{eqnarray}

Now, if $W$ be value of the cut

\begin{eqnarray}
E[W]  & = & \frac {1}{\pi} \sum  w_{ij} \cos^{-1}{(v_i \cdot
v_j)} \nonumber\\
      & \ge & \alpha \cdot Z_P^*
\end{eqnarray}

Note that,

\[
 Z_P^* = \frac {1}{2} \sum w_{ij} \left( 1-v_i \cdot v_j \right)
\]

The inequality of (3) holds, because, we can prove the inequality
term by term, as follows.
\[
\frac {\cos^{-1}{(v_i \cdot v_j)}}{\pi} \ge \alpha \cdot
\frac{1}{2}\left( 1-v_i \cdot v_j \right)
\]

\[
\frac {\theta}{\pi} \ge \alpha \cdot \frac{1}{2}\left( 1-\cos \theta
\right)
\]

It can be shown that for $\alpha$ = 0.878, the above inequality
always holds. More precisely, it can be shown by simple trigonometry
that 0.878 is minimum ratio $\frac{2 \theta}{\pi \left( 1-\cos
\theta \right)}$

It has shown that it is NP-hard to approximate Max-Cut better than
0.94. There is a belief that 0.878 may be the best we can
approximate within polynomial time. This is a randomized algorithm
and its derandomization has also been done.

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