nearly uniform sparsest cut

main.tex

@@ -193,6 +193,15 @@ I think the intuition behind this SDP relaxation is almost the same as \metric{}
 The Lasserre relaxation of SDP automatically satisfies 1 and 2. But I believe there may be some very strange kind of metric that embeds into $\ell_1$ well?
 
 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.
+
+\section{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. 
+
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ref.bib

@@ -334,3 +334,17 @@ series = {SODA '95}
 	year = {2013},
 	pages = {270--279},
 }
+
+@misc{chawla_composition_2023,
+	title = {Composition of nested embeddings with an application to outlier removal},
+	url = {http://arxiv.org/abs/2306.11604},
+	doi = {10.48550/arXiv.2306.11604},
+	urldate = {2025-05-23},
+	publisher = {arXiv},
+	author = {Chawla, Shuchi and Sheridan, Kristin},
+	month = nov,
+	year = {2023},
+	note = {arXiv:2306.11604 [cs]},
+	keywords = {Computer Science - Data Structures and Algorithms},
+	annote = {Comment: 28 pages (including 2 appendices), 5 figures},
+}