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\title{Approximation Algorithms for Sparsest Cut}
\author{Class 4. \zh{丛宇}, 202411081537\\ Major: Theoretical Computer Science}
\date{}

\DeclareMathOperator*{\opt}{OPT}
\DeclareMathOperator*{\len}{len}
\newcommand{\scut}{\textsc{Sparsest Cut}}
\newcommand{\nonuscut}{\textsc{Non-Uniform Sparsest Cut}}
\newcommand{\expansion}{\textsc{Expansion}}

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\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}
% Requirement: The introduction may have the following parts: 
% Establishing the importance (Background), 
% literature review (previous research contributions)
% Gap/the specific problem
% The present study (the research topic, the research purpose  )
% The information should be presented in coherent paragraphs.

\scut{} is a fundamental problem in graph algorithms with connections to various cut related problems.

\begin{problem}[\nonuscut]
The input is a graph $G=(V,E)$ with edge capacities $c:E\to \R_+$ and a set of vertex pairs $\{s_1,t_1\},\dots,\{s_k,t_k\}$ along with demand values $D_1,\dots,D_k\in \R_+$. The goal is to find a cut $\delta(S)$ of $G$ such that $\frac{c(\delta(S))}{\sum_{i:|S\cap \set{s_i,t_i}|=1}D_i}$ is minimized.
\end{problem}

In other words, \nonuscut{} finds the cut that minimizes its capacity divided by the sum of demands of the vertex pairs it separates. There are two important varients of \nonuscut{}. Note that we always consider unordered pair $\{s_i,t_i\}$, i.e., we do not distinguish $\{s_i,t_i\}$ and $\{t_i,s_i\}$.

\scut{} is the uniform version of \nonuscut{}. The demand is 1 for every possible vertex pair $\{s_i,t_i\}$. In this case, we can remove from the input the pairs and demands. The goal becomes to minimize $\frac{c(\delta(S))}{|S||V\setminus S|}$.

\expansion{} further simplifies the objective of \scut{} to $\min_{|S|\leq n/2}\frac{c(\delta(S))}{|S|}$.

\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. 
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.

\subsection{related works}
\scut{} is APX-hard \citep{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\citep{chawla_hardness_2005}. The currently best approximation algorithm has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \citep{arora_osqrtlogn_2010}. Prior to this currently optimal result, there is a long line of research optimizing both the approximation ratio and the complexity, see \citep{arora_expander_2004,leighton_multicommodity_1999}.
There are also works concerning approximating \scut{} on special graph classes such as planar graphs \citep{lee_genus_2010}, graphs with low treewidth \citep{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}.

For an overview of the LP methods for \scut{}, see \citep{sparsest_cut_notes}.

\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 $2^{O(k)}\poly(n)$.

\section{Literature Review}
% Requirement: summarize previous research contributions and identify the gap or the specific problem

\section{The Research Design}
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% 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
  

\section{Time Table}
% Data collection: e.g. During the program and first 6 months after the program (Aug. 2023- May. 2024)
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