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chao.sty
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\RequirePackage[mathletters]{ucs} % allow Unicode in .tex file
\RequirePackage[utf8]{inputenc}
\RequirePackage[charter]{mathdesign} % change fonts
\RequirePackage{XCharter} % change fonts
% \RequirePackage{XCharter} % change fonts
\RequirePackage{berasans, beramono}
\RequirePackage{eucal}
\RequirePackage[nocompress]{cite} % other convenient stuff
% \RequirePackage[nocompress]{cite} % other convenient stuff
\RequirePackage{enumerate}
%\usepackage{pgf,tikz}
\RequirePackage{wrapfig}

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main.tex
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\documentclass[12pt]{article}
\documentclass[11pt]{article}
\usepackage{chao}
\usepackage{natbib}
@ -38,13 +38,21 @@ 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}
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 tree width \citep{gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}.
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}.
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 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

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ref.bib
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@ -168,3 +168,18 @@ keywords = {sparsest cut, hardness of approximation, Directed multicut}
keywords = {Approximation algorithms, Computer science, Costs, Linear programming, Mathematics},
pages = {144--153},
}
@inproceedings{chlamtac_approximating_2010,
address = {Berlin, Heidelberg},
title = {Approximating {Sparsest} {Cut} in {Graphs} of {Bounded} {Treewidth}},
isbn = {978-3-642-15369-3},
doi = {10.1007/978-3-642-15369-3_10},
language = {en},
booktitle = {Approximation, {Randomization}, and {Combinatorial} {Optimization}. {Algorithms} and {Techniques}},
publisher = {Springer},
author = {Chlamtac, Eden and Krauthgamer, Robert and Raghavendra, Prasad},
editor = {Serna, Maria and Shaltiel, Ronen and Jansen, Klaus and Rolim, José},
year = {2010},
keywords = {General Demand, Linear Programming Relaxation, Linear Programming Solution, Tree Decomposition, Vertex Cover},
pages = {124--137},
}