sdp approximation

main.tex

@@ -45,10 +45,10 @@ From a more mathematical perspective, the techniques developed for approximating
 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 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}.
+\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 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}.
 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 \cite{sparsest_cut_notes}.
+For an overview of the LP methods for \scut{}, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}.
 
 % \subsection{open problems}
 
@@ -63,12 +63,14 @@ For \nonuscut{} \cite{leighton_multicommodity_1999} only guarantees a $O(\log^2
 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 \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.
+\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.
 \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\footnote{\url{https://home.ttic.edu/~harry/teaching/teaching.html}}.
 
 
-\section{LP}
+\section{Approximations}
+Techniques for approximating uniform \scut{} and \nonuscut{}.
+\subsection{LP $\Theta(\log n)$}
 
 \begin{minipage}{0.47\linewidth}
 \begin{equation}\label{IP}
@@ -126,7 +128,7 @@ $D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R_+$, $\sum_
 \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}
+\paragraph{$\Theta(\log n)$ flow-cut gap} 
 The flow-cut gap is $\opt(\ip{})/\opt(\lp{})$ \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 $1/\opt(\lp{})$.
@@ -135,11 +137,12 @@ For 1 and 2-commodity flow problem the gap is 1 \cite{Ford_Fulkerson_1956,Hu_196
 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.
 
-\paragraph{$\Theta(\log n)$ flow-cut gap} 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 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 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 theorem \cite{sparsest_cut_notes}. (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?)
+\cite{leighton_multicommodity_1999} gave an algorithmic proof based on \metric{}. This can also be proven using metric embedding theorem, see \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}. (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{}?)
 
-\paragraph{$O(\sqrt{\log n})$ approximation}
+\subsection{SDP $O(\sqrt{\log n})$}
+SDP approximation follows from metric embedding results.
 
 \bibliographystyle{alpha}
 \bibliography{ref}

ref.bib

@@ -1,12 +1,4 @@
 
-@misc{sparsest_cut_notes,
-  author       = {Chekuri, Chandra},
-  title        = {Introduction to Sparsest Cut},
-  howpublished = {Lecture notes, UIUC CS 598CSC: Topics in Graph Algorithms},
-  year         = {2024},
-  note         = {Accessed on May 9, 2025},
-  url          = {https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}
-}
 
 @article{hoory_expander_2006,
   title    = {Expander graphs and their applications},