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main.tex

@@ -136,7 +136,7 @@ For the $\Omega(\log n)$ lowerbound consider an \uscut{} instance on some 3-regu
 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. 
 We can solve \metric{} in polynomial time and get a metric on $V$. Then there is an embedding of $V$ into $\R^d$ with $\ell_1$ metric such that the distortion is $O(\log n)$. 
-Since $\ell_1$ metric is in the cut cone, our metric on $\R^d$ is a conic combination of cut metrics, which implies\footnote{This requires some work. See \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}} that there is a cut in the conic combination with value at most $O(\log n)\opt(\metric{})$. 
+Since $\ell_1$ metric is in the cut cone, our metric on $\R^d$ is a conic combination of cut metrics, which implies\footnote{for details see Thm11 in \url{https://courses.grainger.illinois.edu/cs598csc/fa2024/Notes/lec-sparsest-cut.pdf}} that there is a cut in the conic combination with value at most $O(\log n)\opt(\metric{})$. 
 To find such a cut it suffices to compute a conic combination of cut metrics which is exactly our $\ell_1$ metric in $\R^d$. One way to do this is test $(n-1)d$ cuts by observing the followings,
 \begin{enumerate}
 \item Every coordinate of $\R^d$ corresponds to a line metric;
@@ -150,7 +150,7 @@ The gap can be improved to $\log k$ through a stronger metric embedding theorem
 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 \texorpdfstring{$O(\sqrt{\log n})$}{O(√log n)} - \uscut{}}
+\subsection{SDP \texorpdfstring{$O(\sqrt{\log n})$}{} - \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}.
 
 \begin{equation*}
@@ -197,18 +197,22 @@ The Lasserre relaxation of SDP automatically satisfies 1 and 2. But I believe th
 Another possible approach for \nonuscut{} would be making the number of demand vertices small and then applying a metric embedding (contraction) to $\ell_1$ with better distortion on those vertices.
 \end{remark}
 
-\subsection{SDP \texorpdfstring{$O(\sqrt{\log n}\log \log n)$}{O(√log n log log n)}-\nonuscut}
+\subsection{SDP \texorpdfstring{$O(\sqrt{\log n}\log \log n)$}{} - \nonuscut}
 
 Arora, Lee and Naor \cite{arora_euclidean_2005,arora_frechet_2007} proved that there is an embedding from $\ell_2^2$ to $\ell_1$ with distortion $O(\sqrt{\log n}\log \log n)$. This implies an approximation for \nonuscut{} with the same ratio.
 
-\section{Nealy uniform \scut{}}
-What is the best approximation ratio for \uscut{} instances where almost all demands are uniform. 
+\section{What problem can I work on?}
+
+\subsection{Nealy uniform \scut{}}
+What is the best approximation ratio for \uscut{} instances where almost all demands are uniform?
 More formally, consider a \nonuscut{} instance where only $k$ vertices are associated with demand pairs with $D_i\neq 1$, 
 we want to show that we can approximate nearly uniform \scut{} in polynomial time to ratio $O(\sqrt{\log n}f(k))$, where $f(k)=O(\log \log n)$ when $k\to n$. 
 Let those $k$ non uniform vertices be outliers. 
 \cite{arora_expander_2004} shows that for non-outlier verteices the optimal solution to SDP (a metric) can be embedded into $\ell_1$ with distortion $\sqrt{\log n}$.
 \cite{chawla_composition_2023} is a recent result on getting approximate $(k,c)$-outlier embeddings.
 
+This is not a interesting problem since if arbitary demand is allowed on the non-uniform vertices the approximation can be also arbitary. If only constant demand is allowed, then for any constant $k$ we can obtain the same approximation as \uscut{} by ignoring the non-uniform part. This problem does not have much to do with outlier embeddings.
+
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