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@@ -68,25 +68,56 @@ a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running
 \scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination mentioned in \citep{sparsest_cut_notes} is that shortest path distance in trees is an $\ell_1$ metric. There are works concerning planar graphs and more generally graphs with constant genus.
 \citep{leighton_multicommodity_1999} provided a $\Omega(\log n)$ lowerbound for flow-cut gap for \scut{}. However, it is conjectured that the gap is $O(1)$, while currently the best upperbound is still $O(\sqrt{\log n})$ \citep{rao_small_1999}.
 For graphs with constant genus, \citep{lee_genus_2010} gives a $O(\sqrt{\log g})$ approximation for \scut{}, where $g$ is the genus of the input graph.  For flow-cut gap in planar graphs the techniques are mainly related to metric embedding theory.
-\section{The Research Design}
-% Requirement : Your research design may include exact details of your design and the information should be presented in coherent paragraphs:
-% Example:
-% Research type: e.g. qualitative study, using primary data
-% Sources and other important details 
-% Research methods: e.g. Questionnaire surveys and interviews
-% Possible difficulties/ problems or issues worth considering 
-% Data analysis (the specific data analysis method)
-% e.g. Using SPSS to analyze the survey data and Using NVivo to analyze the interview data (details of the method and reasons for the choice)
-% The significance/ implications of the study
-\paragraph{Research type:} theoretical research
-\paragraph{Possible difficulties:} The technical depth of open problems in \scut{} might be larger than I expected. If I have no idea after thoroughly understanding metric embedding methods and SDP relaxation, I will immediately move to other problems.
 
-\section{Time Table}
-% Data collection: e.g. During the program and first 6 months after the program (Aug. 2023- May. 2024)
-% Data analysis: June 2024- Sept. 2024
 
-understanding existing methods: 2 weeks.\newline
-solving a problem or imporving some approximation: at most 2 months.
+\section{LP}
+
+\begin{minipage}{0.47\linewidth}
+\begin{equation}\label{IP}
+\begin{aligned}
+\min&   &   \frac{\sum_e c_e x_e}{\sum_{i} D_i y_i}&    &   &\\
+s.t.&   &   \sum_{e\in p} x_e&\geq y_i                  &   &\forall \mathcal{P}_{s_i,t_i}, \forall i\\
+    &   &   x_e,y_i&\in \{0,1\}
+\end{aligned}
+\end{equation}
+\end{minipage}
+\begin{minipage}{0.47\linewidth}
+\begin{equation}\label{LP}
+\begin{aligned}
+\min&   &   \sum_e c_e x_e&     &   &\\
+s.t.&   &   \sum_i D_iy_i&=1    &   &\\
+    &   &   \sum_{e\in p} x_e&\geq y_i                  &   &\forall \mathcal{P}_{s_i,t_i}, \forall i\\
+    &   &   x_e,y_i&>0
+\end{aligned}
+\end{equation}
+\end{minipage}
+\bigskip
+
+\begin{minipage}{0.47\linewidth}
+\begin{equation}\label{dual}
+\begin{aligned}
+\max&   &   \lambda&    &   &\\
+s.t.&   &   \sum_{p\in\mathcal{P}_{s_i,t_i}} y_p&\geq \lambda D_i   &   &\forall i\\
+    &   &   \sum_i \sum_{p\in \mathcal{P}_{s_i,t_i}, p\ni e}y_p&\geq c_e    & &\forall e\\
+    &   &   y_p&\geq 0
+\end{aligned}
+\end{equation}
+\end{minipage}
+\begin{minipage}{0.47\linewidth}
+\begin{equation}\label{metric}
+\begin{aligned}
+\min&   &   \sum_{uv\in E} c_{uv}d(u,v)&     &   &\\
+s.t.&   &   \sum_i D_i d(s_i,t_i)&=1    &   &\\
+    &   &   \text{$d$ is a metric on $V$}
+\end{aligned}
+\end{equation}
+\end{minipage}
+
+\newcommand{\lp}{LP\ref{LP}}
+\newcommand{\ip}{IP\ref{IP}}
+\begin{enumerate}
+\item \ip $\geq$ \lp. Given any feasible solution to \ip, we can 
+\end{enumerate}
 
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