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\lecture{22} {Using LP techniques to design algorithms for combinatorial problems }{Aditya Varma}

\section{Integer Linear Programs}
An \emph{Integer Linear Program} is a linear program where the variables are constrained to take integer values only.
An ILP is formulated as 
\begin{center}
max $c^Tx$ \\
$Ax \leq b$ \\
$x_i$ is integral $\forall i$.
\end{center}


\section{Recap of primal,dual and complementary slackness}
Recapitulate that for an LP
\begin{center} 
max $c^Tx$ \\
$Ax \leq b$ \\
\end{center}

the dual is 
\begin{center}
min $y^Tb$ \\
$A^Ty = c$ 
\end{center}

Complementary slackness implies that the optimums of the primal and dual coincide i.e. $c^{T}x_o = (y_o)^Tb$
and where $y_{oi} > 0$ in the dual $A_ix_0 = b_i$ in the primal and vice versa.   

\section{Formulation Of Minimum Spanning Tree as an ILP}
\paragraph
A \emph{spanning tree} is a subgraph of a connected,undirectional graph that connects all vertices and is a tree.
The minimum spanning tree is a spanning tree with minimum cost or sum of weight of edges part of the tree.
The corresponding ILP may be formulated taking  as variables $x_e$ ,one for each edge where a value of $1$ for $x_e$ indicates 
that the edge is part of the tree and value of zero indiacates that the edge is not part of the spanning tree.
The cost then is $\sum x_ec_e$ where $c_e$ is the weight of the edge. 
The ILP formulation then is 
\begin{center}
min $c^Tx = \sum x_e c_e$ \\
\end{center}
\emph{Constraints :} 
\begin{center}
$\forall$ partitions $\pi$  \\
$\displaystyle\sum_{e crosses \pi} x_e \geq  \#(\pi)-1$ \\
$0 \leq x_e \leq 1$ \\
$x_e$ is integral
\end{center}
where $\#(\pi)$ is the number of parts in the partition $\pi$.
The first constraint basically represents that the graph is connected.

Now we drop the integral constraint and write the dual of the resulting LP.
The dual is
\begin{equation*}
\max ( \sum_{\pi} y_{\pi} (\#(\pi) - 1) )
\end{equation*}
\begin{equation*}
\forall e : \displaystyle\sum_{e crosses \pi} y_{\pi} \leq c_e \\
\end{equation*}
\begin{equation*}
y_{\pi} \geq 0 
\end{equation*}
\end{document}