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main.tex

@@ -38,7 +38,7 @@ In other words, \nonuscut{} finds the cut that minimizes its capacity divided by
 
 \expansion{} further simplifies the objective of \scut{} to $\min_{|S|\leq n/2}\frac{c(\delta(S))}{|S|}$.
 
-\subsection{importance and connections}
+% \subsection{importance and connections}
 
 These problems are interesting since they are related to central concepts in graph theory and help to design algorithms for hard problems on graph. One connections is expander graphs. The importance of expander graphs is thoroughly surveyed in \cite{hoory_expander_2006}. The optimum of \expansion{} is also known as Cheeger constant or conductance of a graph. \scut{} provides a 2-approximation of Cheeger constant, which is especially important in the context of expander graphs as it is a way to measure the edge expansion of a graph. \nonuscut{} is related to other cut problems such as Multicut and Balanced Separator.
 From a more mathematical perspective, the techniques developed for approximating \scut{} are deeply related to metric embedding, which is another fundamental problem in geometry.
@@ -50,12 +50,10 @@ There are also works concerning approximating \scut{} on special graph classes s
 
 For an overview of the LP methods for \scut{}, see \cite{sparsest_cut_notes}.
 
-\subsection{open problems}
+% \subsection{open problems}
 
-One major open problem for \scut{} is the best approximation ratio for planar graphs. It is conjectured that the ratio for planar graphs is $O(1)$ but currently the best lowerbound is $O(\sqrt{\log n})$. For graphs treewidth $k$, an open problem is that whether there is a 2 approximation algorithm that runs in $2^{O(k)}\poly(n)$.
+% One major open problem for \scut{} is the best approximation ratio for planar graphs. It is conjectured that the ratio for planar graphs is $O(1)$ but currently the best lowerbound is $O(\sqrt{\log n})$. For graphs treewidth $k$, an open problem is that whether there is a 2 approximation algorithm that runs in $2^{O(k)}\poly(n)$.
 
-\section{Literature Review}
-% Requirement: summarize previous research contributions and identify the gap or the specific problem
 The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap. They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which can imply $O(\log n)$ approximation for \scut{} and $O(\log^2 n)$ approximation for \nonuscut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap.
 
 For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ approximation. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \cite{lognGapAumann98} considered the metric relaxation of the IP. They observed that \nonuscut{} is polynomial time solvable for trees and more generally for all $\ell_1$ metrics. The $O(\log n)$ approximation follows from the $O(\log n)$ distortion in the metric embedding theorem.