fix wrong facts in intro

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@ -10,21 +10,14 @@
\DeclareMathOperator*{\opt}{OPT}
\DeclareMathOperator*{\len}{len}
\newcommand{\scut}{\textsc{Sparsest Cut}}
\newcommand{\uscut}{\textsc{Uniform Sparsest Cut}}
\newcommand{\nonuscut}{\textsc{Non-Uniform Sparsest Cut}}
\newcommand{\expansion}{\textsc{Expansion}}
\begin{document}
\maketitle
% \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.
@ -34,28 +27,34 @@ The input is a graph $G=(V,E)$ with edge capacities $c:E\to \R_+$ and a set of v
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 $\frac{c(\delta(S))}{|S||V\setminus S|}$.
\uscut{} 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 $\min_{|S|\leq n/2}\frac{c(\delta(S))}{|S|}$.
\expansion{} further simplifies the objective of \uscut{} 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 \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.
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. \uscut{} provides a 2-approximation of \expansion{}, 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}
\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \scut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming that NP-complete problems cannot be solved in randomized subexponential time. The currently best approximation algorithm for \scut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{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 \cite{arora_expander_2004,leighton_multicommodity_1999}.
\nonuscut{} is APX-hard \cite{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\cite{chawla_hardness_2005}. \uscut{} admits no PTAS \cite{uniformhardnessFocs07}, assuming that NP-complete problems cannot be solved in randomized subexponential time. The currently best approximation algorithm for \uscut{} has ratio $O(\sqrt{\log n})$ and running time $\tilde{O}(n^2)$ \cite{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 \cite{arora_expander_2004,leighton_multicommodity_1999}.
For \nonuscut{} the best approximation is $O(\sqrt{\log n}\log\log n)$ \cite{arora_euclidean_2005,arora_frechet_2007}.
There are also works concerning approximating \scut{} on special graph classes such as planar graphs \cite{lee_genus_2010}, graphs with low treewidth \cite{chlamtac_approximating_2010,gupta2013sparsestcutboundedtreewidth, Chalermsook_2024}.
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 (in fact the tight $\Theta(\log n)$ gap was proven by \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995}). They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which implies $O(\log n)$ approximation for \scut{}. 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.
The seminal work of \cite{leighton_multicommodity_1988,leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap for \uscut{}.
% (the tight $\Theta(\log n)$ gap for \nonuscut{} was proven by \cite{garg_approximate_1996}).
They developed a $O(\log n)$ approximation algorithm for \uscut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$.
Note that the flow-cut gap describes the ratio of the max concurrent flow to the min sparsity of a cut. \cite{garg_approximate_1996} studied the flow-cut gap for min multicut and max multicommodity flow, which is also $\Theta(\log n)$. The result of Garg, Vazirani and Yannakakis \cite{garg_approximate_1996} provides an $O(\log n)$ approximation algorithm for Multicut, which implies a $O(\log^2 n)$ approximation for \nonuscut{}.
Although \cite{leighton_multicommodity_1999} showed an $\Omega(\log n)$ lowerbound for flow-cut gap, better approximation for \scut{} is still possible through other methods.
For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees an approximation ratio of $O(\log^2 n)$. 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}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} 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.
For \nonuscut{} the $O(\log^2 n)$ approximation is further improved by \cite{Linial_London_Rabinovich_1995} and \cite{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ flow-cut gap as well as a $O(\log n)$ approximation algorithm for \nonuscut{}. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \cite{lognGapAumann98}, \cite{aumann_rabani_1995} and \cite{Linial_London_Rabinovich_1995} 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 $O(\log n)$ gap follows from the $O(\log n)$ distortion in the metric embedding theorem.
\cite{arora_expander_2004} and \cite{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{}.
\cite{arora_expander_2004} and \cite{arora_osqrtlogn_2010} further improved the approximation ratio for \uscut{} to $O(\sqrt{\log n})$ via semidefinite relaxation. This is currently the best approximation ratio for \uscut{} on general undirected graphs.
For \nonuscut{}, the approximation is improved to $O(\sqrt{\log n} \log \log n)$ \cite{arora_euclidean_2005,arora_frechet_2007}. Later \cite{guruswami_approximating_2013} gives a $\frac{1+\delta}{\e}$ approximation in time $2^{r/(\delta \e)}\poly(n)$ provided that $\lambda_r\geq \opt / (1-\delta)$.
There is also plenty of research concerning \scut{} on some graph classes, for example \cite{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \cite{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \cite{Cohen-Addad_Mömke_Verdugo_2024} obtained
There is also plenty of research on \scut{} in some graph classes, for example \cite{bonsma_complexity_2012}. One of the most popular class is graphs with constant treewidth. \cite{Chalermsook_2024} gave a $O(k^2)$ approximation algorithm with complexity $2^{O(k)}\poly(n)$. \cite{Cohen-Addad_Mömke_Verdugo_2024} obtained
a 2-approximation algorithm for sparsest cut in treewidth $k$ graph with running time $2^{2^{O(k)}}\poly(n)$.
\scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination\footnote{\url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}} 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.
@ -64,8 +63,8 @@ For graphs with constant genus, \cite{lee_genus_2010} gives a $O(\sqrt{\log g})$
\section{Approximations}
Techniques for approximating uniform \scut{} and \nonuscut{}.
\subsection{LP $\Theta(\log n)$}
Techniques for approximating \scut{}.
\subsection{LP $\Theta(\log n)$ - \nonuscut{}}
\begin{minipage}{0.47\linewidth}
\begin{equation}\label{IP}
@ -124,14 +123,14 @@ $D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R_+$, $\sum_
Note that $D$ is routable iff the optimum of the LPs is at least 1. Then the theorem follows directly from \metric{}.
\paragraph{$\Theta(\log n)$ flow-cut gap}
The flow-cut gap is $\opt(\ip{})/\opt(\lp{})$ \cite{leighton_multicommodity_1999}.
The flow-cut gap is defined as $\opt(\ip{})/\opt(\lp{})$ and the $\Theta(\log n)$ bound is proven in \cite{leighton_multicommodity_1999}.
Suppose that $G$ satisfies the cut condition, that is, $c(\delta(S))$ is at least the demand separated by $\delta(S)$ for all $S\subset V$. This implies $\opt(\ip{})\geq 1$ and in this case the largest integrality gap is $1/\opt(\lp{})$.
For 1 and 2-commodity flow problem the gap is 1 \cite{Ford_Fulkerson_1956,Hu_1963}.
However, for $k\geq 3$ the gap becomes larger\footnote{\url{https://en.wikipedia.org/wiki/Approximate_max-flow_min-cut_theorem}}.
It is mentioned in \cite{leighton_multicommodity_1999} that \cite{schrijver_homotopic_1990} proved if the demand graph does not contain either three disjoint edges or a triangle and a disjoint edge, then the gap is 1.
For the $\Omega(\log n)$ lowerbound consider an uniform \scut{} instance on some 3-regular graph $G$ with unit capacity. In \cite{leighton_multicommodity_1999} they further required that for any $S\subset V$ and small constant $c$, $|\delta(S)|\geq c \min(|S|,|\bar S|)$. Then the value of the sparsest cut is at least $\frac{c}{n-1}$. Observe that for any fixed vertex $v$, there are at most $n/2$ vertices within distance $\log n-3$ of $v$. Thus at least half of the $\binom{n}{2}$ demand pairs are connected with shortest path of length at least $\log n-2$. To sustain a flow $f$ we need at least $\frac{1}{2}\binom{n}{2}(\log n -2)f\leq 3n/2$. Any feasible flow satisfies $f\leq \frac{3n}{\binom{n}{2}(\log n -2)}$ and the gap is therefore $\Omega(\log n)$.
For the $\Omega(\log n)$ lowerbound consider an \uscut{} instance on some 3-regular graph $G$ with unit capacity. In \cite{leighton_multicommodity_1999} they further required that for any $S\subset V$ and small constant $c$, $|\delta(S)|\geq c \min(|S|,|\bar S|)$. Then the value of the sparsest cut is at least $\frac{c}{n-1}$. Observe that for any fixed vertex $v$, there are at most $n/2$ vertices within distance $\log n-3$ of $v$. Thus at least half of the $\binom{n}{2}$ demand pairs are connected with shortest path of length at least $\log n-2$. To sustain a flow $f$ we need at least $\frac{1}{2}\binom{n}{2}(\log n -2)f\leq 3n/2$. Any feasible flow satisfies $f\leq \frac{3n}{\binom{n}{2}(\log n -2)}$ and the gap is therefore $\Omega(\log n)$.
For the upperbound it suffices to show there exists a cut of ratio $O(f\log n)$.
\cite{leighton_multicommodity_1999} gave an algorithmic proof based on \metric{}. This can also be proven using metric embedding results.
@ -144,12 +143,14 @@ To find such a cut it suffices to compute a conic combination of cut metrics whi
\item Every line metric on $n$ points can be represented as some conic combination of $n-1$ cut metrics.
\end{enumerate}
The gap can be improved to $\log k$ through a stronger metric embedding theorem ($k$ is the number of demand pairs).
\begin{remark}
I believe the later method is more general and works for \nonuscut{}, while the former method is limited to uniform \scut{}. However, the proof in \cite{leighton_multicommodity_1999} may have connections with the proof of Bourgain's thm? why does the method in \cite{leighton_multicommodity_1999} fail to work on \nonuscut{}?
I believe the later method is more general and works for \nonuscut{}, while the former method is limited to \uscut{}. However, the proof in \cite{leighton_multicommodity_1999} may have connections with the proof of Bourgain's thm? Why does their method fail to work on \nonuscut{}?
\end{remark}
\subsection{SDP $O(\sqrt{\log n})$}
SDP approximation follows from metric embedding results.
\subsection{SDP $O(\sqrt{\log n})$ - \uscut{}}
This $O(\sqrt{\log n})$ approximation via SDP is developed in \cite{arora_expander_2004}. This is also described in \cite[section 15.4]{Williamson_Shmoys_2011}.
\bibliographystyle{alpha}
\bibliography{ref}

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@ -73,6 +73,33 @@
year = {1999},
pages = {787--832}
}
@INPROCEEDINGS{leighton_multicommodity_1988,
author={Leighton, T. and Rao, S.},
booktitle={[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science},
title={An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms},
year={1988},
volume={},
number={},
pages={422-431},
keywords={Application software;Approximation algorithms;Laboratories;Computer science;Particle separators;Linear programming;Contracts;Mathematics;Constraint theory;Bifurcation},
doi={10.1109/SFCS.1988.21958}}
@article{garg_approximate_1996,
title = {Approximate {Max}-{Flow} {Min}-({Multi}){Cut} {Theorems} and {Their} {Applications}},
volume = {25},
issn = {0097-5397},
url = {https://epubs.siam.org/doi/10.1137/S0097539793243016},
doi = {10.1137/S0097539793243016},
number = {2},
urldate = {2025-05-20},
journal = {SIAM Journal on Computing},
author = {Garg, Naveen and Vazirani, Vijay V. and Yannakakis, Mihalis},
month = apr,
year = {1996},
note = {Publisher: Society for Industrial and Applied Mathematics},
pages = {235--251},
}
@inproceedings{lee_genus_2010,
title = {Genus and the geometry of the cut graph: [extended abstract]},
@ -289,4 +316,5 @@ series = {SODA '95}
month = jan,
year = {2013},
pages = {295--305},
}
}
@book{Williamson_Shmoys_2011, place={Cambridge}, title={The Design of Approximation Algorithms}, publisher={Cambridge University Press}, author={Williamson, David P. and Shmoys, David B.}, year={2011}}