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how to show that y_i is the distance...?

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@ -19,8 +19,8 @@
% \paragraph{Research problem} What is the best possible approximation rate of linear programming based approximation algorithms for \scut? What about algorithms for planar graphs? % \paragraph{Research problem} What is the best possible approximation rate of linear programming based approximation algorithms for \scut? What about algorithms for planar graphs?
\section{Introduction} \section{Introduction}
% Requirement: The introduction may have the following parts: % Requirement: The introduction may have the following parts:
% Establishing the importance (Background), % Establishing the importance (Background),
% literature review (previous research contributions) % literature review (previous research contributions)
% Gap/the specific problem % Gap/the specific problem
% The present study (the research topic, the research purpose ) % The present study (the research topic, the research purpose )
@ -40,7 +40,7 @@ In other words, \nonuscut{} finds the cut that minimizes its capacity divided by
\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 \citep{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. 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 \citep{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. From a more mathematical perspective, the techniques developed for approximating \scut{} are deeply related to metric embedding, which is another fundamental problem in geometry.
Besides theoretical interests, \scut{} is useful in practical scenarios such as in image segmentation and in some machine leaning algorithms. Besides theoretical interests, \scut{} is useful in practical scenarios such as in image segmentation and in some machine leaning algorithms.
@ -63,7 +63,7 @@ For \nonuscut{} \citep{leighton_multicommodity_1999} only guarantees a $O(\log^2
\citep{arora_expander_2004} and \citep{arora_osqrtlogn_2010} further improved the approximation ratio for \scut{} to $O(\sqrt{\log n})$ via semidefinite relaxation. This is currently the best approximation ratio for \scut{}. \citep{arora_expander_2004} and \citep{arora_osqrtlogn_2010} further improved the approximation ratio for \scut{} to $O(\sqrt{\log n})$ via semidefinite relaxation. This is currently the best approximation ratio for \scut{}.
There is also plenty of research concerning \scut{} on some graph classes, for example \citep{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \citep{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \citep{Cohen-Addad_Mömke_Verdugo_2024} obtained There is also plenty of research concerning \scut{} on some graph classes, for example \citep{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \citep{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \citep{Cohen-Addad_Mömke_Verdugo_2024} obtained
a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$. a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$.
\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. \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}. \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}.
@ -120,7 +120,7 @@ s.t.& & \sum_i D_i d(s_i,t_i)&=1 & &\\
\begin{enumerate} \begin{enumerate}
\item \ip{} $\geq$ \lp{}. Given any feasible solution to \ip{}, we can scale all $x_e$ and $y_i$ simultaneously with factor $1/\sum_i D_i y_i$. The scaled solution is feasible for \lp{} and gets the same objective value. \item \ip{} $\geq$ \lp{}. Given any feasible solution to \ip{}, we can scale all $x_e$ and $y_i$ simultaneously with factor $1/\sum_i D_i y_i$. The scaled solution is feasible for \lp{} and gets the same objective value.
\item \lp{} $=$ \dual{}. by duality. \item \lp{} $=$ \dual{}. by duality.
\item \metric{} $=$ \lp{}. It is easy to see \metric{} $\geq$ \lp{} since any feasible metric to \metric{} induces a feasible solution to \lp{}. In fact, any solution to \lp{} also induces a feasible metric. \item \metric{} $=$ \lp{}. It is easy to see \metric{} $\geq$ \lp{} since any feasible metric to \metric{} induces a feasible solution to \lp{}. In fact, the optimal solution to \lp{} also induces a feasible metric. Consider a solution $x_e,y_i$ to \lp{}. Let $d$ be the shortest path metric on $V$ using edge length $x_e$. It suffices to show that $y_i$ is the shortest path distance fron $s_i$ to $t_i$. Suppose $y_i\leq d(s_i,t_i)$...
\end{enumerate} \end{enumerate}
\bibliographystyle{plainnat} \bibliographystyle{plainnat}