remove natbib. some flow-cut gap

main.tex

@@ -1,6 +1,6 @@
 \documentclass[11pt]{article}
 \usepackage{chao}
-\usepackage{natbib}
+% \usepackage{natbib}
 
 
 \title{Sparsest Cut}
@@ -40,15 +40,15 @@ In other words, \nonuscut{} finds the cut that minimizes its capacity divided by
 
 \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 \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.
 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 \citep{juliaJACMapxhard} and, assuming the Unique Game Conjecture, has no polynomial time constant factor aproximation algorithm\citep{chawla_hardness_2005}. \scut{} admits no PTAS \citep{uniformhardnessFocs07}, assuming a widely believed conjecture. 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}.
+\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 a widely believed conjecture. The currently best approximation algorithm 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}.
+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}.
 
-For an overview of the LP methods for \scut{}, see \citep{sparsest_cut_notes}.
+For an overview of the LP methods for \scut{}, see \cite{sparsest_cut_notes}.
 
 \subsection{open problems}
 
@@ -58,16 +58,16 @@ One major open problem for \scut{} is the best approximation ratio for planar gr
 % 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{} \citep{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \citep{Linial_London_Rabinovich_1995} and \citep{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. \citeauthor{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.
+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.
 
-\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{}.
+\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{}.
 
-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 \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 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}.
-For graphs with constant genus, \citep{lee_genus_2010} gives a $O(\sqrt{\log g})$ approximation for \scut{}, where $g$ is the genus of the input graph.  For flow-cut gap in planar graphs the techniques are mainly related to metric embedding theory.
+\scut{} is easy on trees and the flow-cut gap is 1 for trees. One explaination mentioned in \cite{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.
+\cite{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 $O(\sqrt{\log n})$ \cite{rao_small_1999}.
+For graphs with constant genus, \cite{lee_genus_2010} gives a $O(\sqrt{\log g})$ approximation for \scut{}, where $g$ is the genus of the input graph.  For flow-cut gap in planar graphs the techniques are mainly related to metric embedding theory.
 
 
 \section{LP}
@@ -124,12 +124,19 @@ s.t.&   &   \sum_i D_i d(s_i,t_i)&=1    &   &\\
 
 \end{enumerate}
 \begin{theorem}[Japanese Theorem]
-$D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R^+$, $\sum_e c_e l(e)\geq \sum_{uv} D(u,v) d_l(u,v)$, where $d_l(s,t)$ is the short path distance induced by $l(e)$.
+$D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R_+$, $\sum_e c_e l(e)\geq \sum_{uv} D(u,v) d_l(u,v)$, where $d_l(s,t)$ is the short path distance induced by $l(e)$.
 \end{theorem}
 Note that $D$ is routable iff the optimum of the LPs is at least 1. Then the theorem follows directly from \metric{}.
 
 \subsection{Flow-cut gap}
+The flow-cut gap is $\opt(\ip{})/\opt(\lp{})$.
 
-\bibliographystyle{plainnat}
+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 $1/\opt(\lp{})$.
+Note that the cut condition is weaker than the routable condition and it is always satisfiable via scaling numbers.
+
+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}}.
+
+\bibliographystyle{alpha}
 \bibliography{ref}
 \end{document}

ref.bib

@@ -225,3 +225,7 @@
 	year = {1999},
 	pages = {300--306},
 }
+
+@article{Ford_Fulkerson_1956, title={Maximal Flow Through a Network}, volume={8}, rights={https://www.cambridge.org/core/terms}, ISSN={0008-414X, 1496-4279}, url={https://www.cambridge.org/core/product/identifier/S0008414X00036890/type/journal_article}, DOI={10.4153/CJM-1956-045-5}, journal={Canadian Journal of Mathematics}, author={Ford, L. R. and Fulkerson, D. R.}, year={1956}, pages={399–404}, language={en} }
+
+@article{Hu_1963, title={Multi-Commodity Network Flows}, volume={11}, ISSN={0030-364X, 1526-5463}, url={https://pubsonline.informs.org/doi/10.1287/opre.11.3.344}, DOI={10.1287/opre.11.3.344}, number={3}, journal={Operations Research}, author={Hu, T. C.}, year={1963}, month=jun, pages={344–360}, language={en} }