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\lecture{22.}{ Using LP techniques to design algorithms for combinatorial problems : Deriving Kruskal's MST algorithm using LP duality}{Manjinder Singh Chauhan}

\section{Integer Linear Program(ILP)}
An Integer linear program is a linear program with an extra constraint that all $x_i$ are integral.\\
Formally
\begin{gather}
\text{max} \enskip c^{T}x \notag\\
\text Ax  \leq b \notag 
\end{gather}
\\
where $x_i$ is intergral $\forall i$ \notag

\paragraph
  {Now we allready know that that for}

\begin{center}
\begin{tabular}{r|l}
\underline{Primal} & \underline{Dual} \\
 & \\
max $c^Tx$   &    min $y^Tb$\\
$Ax \leq b$ & $A^Ty = c$ \\
 & $y \geq 0$ \\
\end{tabular}
\end{center}

if x, y are feasiable and\\
 $y_i > 0 $  \Longrightarrow $A_i$x=$b_i$\\ 
 then the solution is optimal.



\paragraph
 {The second question in quiz 2 was an attempt to make the above process iterative.}

\begin{center}
\begin{tabular}{r|l}
 & \\
max $c^Tx$   &    max $c^Ty$\\
$Ax \leq b$ & $A_1 y \leq 0 $ \\
 for an $x_o$ &  \\
$A_1 x_o = b_1$ &This can be solved much more easily\\
$A_2 x_o < b_2$ &\\
\end{tabular}
\end{center}

\\
Now $x=x_o + \epsilon y$ \\
where $ \epsilon $ depends on A.\\

\section { Formulation of Minimum Spanning Tree as an ILP}

Variables: $X_e$ , one per edge \\
Cost: {min} \enskip C^{T}X \\
  $where C_e$ is cost of edge e\\
Constraint: $0 \leq X_e \leq 1 $  and $X_e$ is an integer\\
\\
\\
\begin{center}
$\forall$  partition  $\pi $\\\\
\sum_{e crosses \pi} X_e \geq  \#(\pi)-1 \\
where $\#(\pi) $is the number of parts in a partition. \\
\end{center}

\paragraph
{Next we drop the constraint that $X_e$ is an integer and also that $X_e \leq 1$} 
\\
Now writing the dual of the MST linear program we get \\
\begin{eqnarray}
{max} \sum_{\pi} $y_\pi (\#(\pi) -1)$ \notag \\
$\forall $ e   \sum_{e crosses \pi}  $y_{\pi} \leq C_e $ \notag\\
 $y_{\pi} \geq 0 $  \notag

\end{eqnarray}



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