use action

.gitea/workflows/compile.yml

@@ -0,0 +1,29 @@
+name: build pdf
+on: [push,watch]
+
+jobs:
+  build:
+    runs-on: macos  # use label
+
+    steps:
+      - name: Check out the repository
+        uses: actions/checkout@v4
+
+      - name: Compile LaTeX using local TeX Live
+        # These commands run directly in your machine's shell
+        run: |
+          echo "Compiling document..."
+          latexmk -pdf main.tex
+      
+      - name: List files in the workspace
+        run: ls -l
+
+      - uses: akkuman/gitea-release-action@v1
+        with:
+          body: ''
+          prerelease: true
+          name: PDF
+          token: ${{ secrets.RELEASE_TOKEN }}
+          tag_name: latest
+          files: |-
+            ./*.pdf

.gitignore

@@ -1,6 +1,7 @@
 
 .vscode/
 *.sage.py
+/*.pdf
 ## Core latex/pdflatex auxiliary files:
 *.aux
 *.lof

algo.sty

@@ -0,0 +1,14 @@
+\def\begin@lg{\begin{minipage}{1in}\begin{tabbing}
+        \quad\=\qquad\=\qquad\=\qquad\=\qquad\=\qquad\=\qquad\=\kill}
+\def\end@lg{\end{tabbing}\end{minipage}}
+
+\newenvironment{algorithm}
+{\begin{tabular}{|l|}\hline\begin@lg}
+{\end@lg\\\hline\end{tabular}}
+
+\newenvironment{algo}
+{\begin{center}\begin{algorithm}}
+{\end{algorithm}\end{center}}
+
+\def\argmax{\operatornamewithlimits{arg\,max}}
+\def\argmin{\operatornamewithlimits{arg\,min}}

latexmkrc

@@ -0,0 +1 @@
+$pdflatex=q/xelatex -synctex=1 %O %S/

main.tex

@@ -1,5 +1,6 @@
 \documentclass[11pt]{article}
-\usepackage{chao}
+% \usepackage{chao}
+\usepackage[sans]{myctex}
 % \usepackage{natbib}
 
 
@@ -64,7 +65,7 @@ For graphs with constant genus, \cite{lee_genus_2010} gives a $O(\sqrt{\log g})$
 
 \section{Approximations}
 Techniques for approximating \scut{}.
-\subsection{LP $\Theta(\log n)$ - \nonuscut{}}
+\subsection{LP \texorpdfstring{$\Theta(\log n)$}{θ(log n)} - \nonuscut{}}
 
 \begin{minipage}{0.47\linewidth}
 \begin{equation}\label{IP}
@@ -135,7 +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;
@@ -149,9 +150,71 @@ 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 $O(\sqrt{\log n})$ - \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*}
+\begin{aligned}
+\min&   &   \frac{\sum_{ij\in E}c_{ij}(x_i-x_j)^2}{\sum_{ij\in V\times V}(x_i-x_j)^2}&  &   &\\
+s.t.&   &   (x_i-x_j)^2 + (x_j-x_k)^2&\geq (x_i-x_k)^2  &   &\forall i,j,k\in V\\
+    &   &                   x_i&\in \{+1,-1\}   &   &\forall i \in V
+\end{aligned}
+\end{equation*}
+
+This SDP models \uscut{} since every assignment of $x$ corresponds to a cut and the objective is the sparsity of the cut (up to a constant factor, but we don't care since we cannot achieve a constant factor approximation anyway). Consider a relaxation which is similar to \lp{}.
+
+\begin{equation*}
+\begin{aligned}
+\min&   &   \sum_{ij\in E}c_{ij}\|v_i-v_j\|^2&  &   &\\
+s.t.&   &                   \sum_{ij\in V\times V}\|v_i-v_j\|^2&=1  &   &\\
+    &   &   \|v_i-v_j\|^2 + \|v_j-v_k\|^2&\geq \|v_i-v_k\|^2  &   &\forall i,j,k\in V\\
+    &   &                   v_i&\in \R^n   &   &\forall i \in V
+\end{aligned}
+\end{equation*}
+
+To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get a cut $\delta(S)$ with $c(\delta(S))=|S| \opt(SDP) O(n\sqrt{\log n})$. If there are two sets $S,T\subset V$ both of size $\Omega(n)$ that are well-separated, in the sense that for any $s\in S$ and $t\in T$, $\|v_s-v_t\|^2=\Omega(1/\sqrt{\log n})$, then the SDP gap follows from
+
+\[
+\frac{c(\delta(S))}{|S||V-S|}
+\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2}
+\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n^2} O(\sqrt{\log n}) 
+\leq O(\sqrt{\log n}) \opt(SDP).
+\]
+
+This is the framework of the proof in \cite{arora_expander_2004}.
+I think the intuition behind this SDP relaxation is almost the same as \metric{}. $\ell_1$ metrics are good since they are in the cut cone. However, if we further require that the metric in \metric{} is an $\ell_1$ metric in $\R^d$, then resulting LP is NP-hard, since the integrality gap becomes 1.
+\cite{leighton_multicommodity_1999} showed that the $\Theta(\log n)$ gap is tight for \metric{}, but add extra constraints to \metric{} (while keeping it to be a relaxation of \scut{} and to be polynomially solvable) may provides better gap. The SDP relaxation is in fact trying to enforce the metric to be $\ell_2^2$ in $\R^n$.
+
+\begin{remark}
+$O(\sqrt{\log n})$ is likely to be the optimal bound for the above SDP. To get better gap one can stay with SDP and add more additional constraints (like Sherali-Adams, Lovász-Schrijver and Lasserre relaxations); or think distance as variables in an LP and force feasible solution to be certain kind of metrics. \cite{arora_towards_2013} is following the former method and considers Lasserre relaxations. For the later method, getting a cut from the optimal metric is the same as embedding it to $\ell_1$. Thus it still relies on progress in metric embedding theory. Note that both methods need to satisfy
+\begin{enumerate}
+\item the further constrained programs is polynomially solvable,
+\item it remains a relaxation of \scut{},
+\item the gap is better.
+\end{enumerate}
+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.
+\end{remark}
+
+\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.
+
+Recently the $O(\sqrt{\log n}\log \log n)$ gap has been improved to $\Theta(\sqrt{\log n})$\footnote{STOC '25 \url{https://web.math.princeton.edu/~naor/homepage files/local-growth-STOC.pdf}}.
+
+\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.
+
 \bibliographystyle{alpha}
 \bibliography{ref}
 \end{document}

myctex.sty

@@ -0,0 +1,492 @@
+\ProvidesPackage{myctex}
+\RequirePackage[fontset=none, scheme=plain]{ctex}
+\RequirePackage{fontspec}
+
+\newif\ifmyctex@sans
+\myctex@sansfalse % Default is serif
+
+% Declare and process options
+\DeclareOption{sans}{\myctex@sanstrue}
+\DeclareOption*{\PackageWarning{myctex}{Unknown option `\CurrentOption'}}
+\ProcessOptions\relax
+
+% Now use the conditional
+\ifmyctex@sans
+\setCJKmainfont{Source Han Sans SC}[BoldFont=*, ItalicFont=*]
+\setCJKsansfont{Source Han Sans SC}
+\setCJKmonofont{Source Han Sans SC}
+\RequirePackage[fira]{fontsetup}
+\RequirePackage{FiraMono}
+\setmathfont{Fira Math}[range=\setminus]
+
+\else
+\setCJKmainfont{Source Han Serif SC}[BoldFont=*, ItalicFont=*]
+\setCJKsansfont{Source Han Serif SC}
+\setCJKmonofont{Source Han Serif SC}
+\RequirePackage[charter]{mathdesign} % math font only for serif mode
+\fi
+
+% \setCJKmainfont{SourceHanSerifSC-Regular.otf}[BoldFont={SourceHanSerifSC-Bold.otf}, ItalicFont=SourceHanSerifSC-Regular.otf]
+
+\RequirePackage[margin=1in]{geometry}
+\RequirePackage{latexsym,amsmath}
+\RequirePackage[dvipsnames]{xcolor}
+
+\RequirePackage[colorlinks=true,urlcolor=Blue,citecolor=Green,linkcolor=BrickRed,unicode]{hyperref}
+
+\RequirePackage[hyperref]{ntheorem}   % theorem numbering
+\RequirePackage{aliascnt}
+\RequirePackage{xspace}
+% \RequirePackage{minted}
+
+
+\RequirePackage{graphicx} % essentials
+% \RequirePackage{microtype}
+\RequirePackage{eucal}
+\RequirePackage[nocompress]{cite}       % other convenient stuff
+\RequirePackage{enumerate}
+%\usepackage{pgf,tikz}
+\RequirePackage{wrapfig}
+\RequirePackage[medium,compact]{titlesec}
+\RequirePackage{verbatim}
+\newtheorem{theorem}{Theorem}[section]
+
+\newaliascnt{lemma}{theorem}
+\newaliascnt{corollary}{theorem}
+\newaliascnt{conjecture}{theorem}
+\newaliascnt{proposition}{theorem}
+
+\newtheorem{lemma}{Lemma}[section]
+\newtheorem{corollary}{Corollary}[section]
+\newtheorem{conjecture}{Conjecture}[section]
+\newtheorem{proposition}{Proposition}[section]
+\newtheorem{problem}{Problem}
+
+\newcommand{\lemmaautorefname}{Lemma}
+\newcommand{\corollaryautorefname}{Corollary}
+\newcommand{\conjectureautorefname}{Conjecture}
+\newcommand{\propositionautorefname}{Proposition}
+\newcommand{\problemautorefname}{Problem}
+
+\newenvironment{definition}[1][Definition]{\begin{trivlist}
+\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
+\newenvironment{example}[1][Example]{\begin{trivlist}
+\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
+\newenvironment{remark}[1][Remark]{\begin{trivlist}
+\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
+
+\widowpenalty 5000
+\clubpenalty 5000
+
+\numberwithin{figure}{section}
+\pagenumbering{arabic}
+
+\urlstyle{same}     % dammit
+
+\hyphenation{co-or-din-ate co-or-din-ates half-plane half-space stereo-iso-mers stereo-iso-mer Round-table homol-ogous homol-ogy poly-hedron poly-hedra homeo-morphic homeo-morph-ism}
+
+\pdfstringdefDisableCommands{\let\boldmath\relax} % allow \boldmath in section titles
+
+\let\subsectionautorefname\sectionautorefname
+\let\subsubsectionautorefname\sectionautorefname
+
+
+% ----------------------------------------------------------------------
+%  Common Latin abbreviations, only one of which is actually kosher
+% ----------------------------------------------------------------------
+
+\def\etal{\emph{et~al.}}            % and others
+\def\ie{\emph{i.e.}}           % that is
+% \def\ie{\textcolor{red}{that is}}
+\def\eg{\emph{e.g.}}           % for example
+% \def\eg{\textcolor{red}{for example}}
+
+% -----------------------
+% Common Names
+% -----------------------
+\def\Mobius{M\"obius\xspace}
+
+% -----------------------
+% useful shortcuts
+% -----------------------
+
+% \def\cost{\cent}
+
+% ----------------------------------------------------------------------
+%  Simple math stuff
+% ----------------------------------------------------------------------
+% \renewcommand{\subset}{\subseteq}
+% ---- SYMBOLS ----
+\let\e\epsilon               % a ``real'' epsilon — better yet, just use Unicode ε.
+%
+%  I give up.  These are in the wrong font, but my kludged versions 
+%  LOOK like kludges, especially \Z, \Q, and \C.
+%
+\def\Real{\mathbb{R}}
+\def\Proj{\mathbb{P}}
+\def\Hyper{\mathbb{H}}
+\def\Integer{\mathbb{Z}}
+\def\Natural{\mathbb{N}}
+\def\Complex{\mathbb{C}}
+\def\Rational{\mathbb{Q}}
+
+\let\N\Natural
+\let\Q\Rational
+\let\R\Real
+\let\Z\Integer
+\def\Rd{\Real^d}
+\def\RP{\Real\Proj}
+\def\CP{\Complex\Proj}
+
+
+
+% ---- OPERATORS (requires amsmath) ----
+\def\aff{\operatorname{aff}}        
+\def\area{\operatorname{area}}
+\def\argmax{\operatornamewithlimits{arg\,max}}
+\def\argmin{\operatornamewithlimits{arg\,min}}
+\def\Aut{\operatorname{Aut}}        % Automorphism group
+\def\card{\operatorname{card}}  % cardinality, deprecated for \abs
+\def\conv{\operatorname{conv}}  
+\def\E{\operatorname{E}}            % Expectation: $\E[X]$ (like \Pr)
+\def\EE{\operatornamewithlimits{E}}
+\def\Hom{\operatorname{Hom}}        % Homomorphism group
+\def\id{\operatorname{id}}      % identity
+\def\im{\operatorname{im}}      % image
+\def\lcm{\operatorname{lcm}}
+\def\lfs{\operatorname{lfs}}        % local feature size
+\def\poly{\operatorname{poly}}
+\def\polylog{\operatorname{polylog}}
+\def\rank{\operatorname{rank}}
+\def\rel{\operatorname{rel\,}}  % relative (interior, boundary, etc.)
+\def\sgn{\operatorname{sgn}}
+\def\vol{\operatorname{vol}}        % volume
+
+\def\fp#1{^{\underline{#1}}}        % falling powers: $n\fp{d}$
+\def\rp#1{^{\overline{#1}}}     % rising powers:  $n\rp{d}$
+
+\def\setsymdiff{\operatorname{\triangle}}
+% --- Darts and fences ---
+% less nice replacements for stmaryrd characters
+\@ifundefined{shortrightarrow}{\let\shortrightarrow\rightarrow}{}
+\@ifundefined{shortleftarrow}{\let\shortleftarrow\leftarrow}{}
+\@ifundefined{shortuparrow}{\let\shortuparrow\uparrow}{}
+\@ifundefined{shortdownarrow}{\let\shortdownarrow\downarrow}{}
+
+\def\arcto{\mathord\shortrightarrow}
+\def\arcfrom{\mathord\shortleftarrow}
+\def\arc#1#2{#1\arcto#2}
+\def\cra#1#2{#1\mathord\shortleftarrow#2}
+\def\fence#1#2{#1\mathord\shortuparrow#2}
+\def\ecnef#1#2{#1\mathord\shortdownarrow#2}
+
+% --- Cheap displaystyle operators ---
+\def\Frac#1#2{{\displaystyle\frac{#1}{#2}}}
+\def\Sum{\sum\limits}
+\def\Prod{\prod\limits}
+\def\Union{\bigcup\limits}
+\def\Inter{\bigcap\limits}
+\def\Lor{\bigvee\limits}
+\def\Land{\bigwedge\limits}
+\def\Lim{\lim\limits}
+\def\Max{\max\limits}
+\def\Min{\min\limits}
+
+% ---- RELATORS ----
+\def\deq{\stackrel{\scriptscriptstyle\triangle}{=}} % Use := instead.
+\def\into{\DOTSB\hookrightarrow}        % = one-to-one
+\def\onto{\DOTSB\twoheadrightarrow}
+\def\inonto{\DOTSB\lhook\joinrel\twoheadrightarrow}
+\def\from{\leftarrow}
+\def\tofrom{\leftrightarrow}
+\def\mapsfrom{\mathrel{\reflectbox{$\mapsto$}}}
+\def\longmapsfrom{\mathrel{\reflectbox{$\longmapsto$}}}
+
+% ---- DELIMITER PAIRS ----
+% --- always self-scaling delmiter pairs ---
+\def\set#1{\left\{ #1 \right\}}
+\def\floor#1{\left\lfloor #1 \right\rfloor}
+\def\ceil#1{\left\lceil #1 \right\rceil}
+\def\seq#1{\left\langle #1 \right\rangle}
+\def\abs#1{\left| #1 \right|}
+\def\norm#1{\left\| #1 \right\|}
+\def\paren#1{\left( #1 \right)}     % need better macro name!
+\def\brack#1{\left[ #1 \right]}     % need better macro name!
+\def\indic#1{\left[ #1 \right]}     % indicator variable; Iverson notation
+
+%
+%  Macros to typeset sets like {foo|bar} with all three delimiters
+%  correctly scaled to fit.  What I *really* want is a \middle macro
+%  that acts just like \left and \right.  Grumble.
+%
+%  Set version is handled by package braket.
+\def\Bigbar#1{\mathrel{\left|\vphantom{#1}\right.\n@space}}
+\def\Seqbar#1#2{\Seq{#1 \Bigbar{#1 #2} #2}}
+\def\Brackbar#1#2{\Brack{#1 \Bigbar{#1 #2} #2}}
+
+%  Math mode fbox
+\def\mfbox#1{\mathchoice{{\fbox{\ensuremath{\displaystyle #1}}}}
+            {{\fbox{\ensuremath{\textstyle #1}}}}
+            {{\fbox{\ensuremath{\scriptstyle #1}}}}
+            {{\fbox{\ensuremath{\scriptscriptstyle #1}}}}}
+
+% ---- ``ACCENTS'' ----
+%   NB: Commands equivalent to \lrarrowfill and \overlrarrow are
+%       already defined in amstex!
+\def\lrarrowfill{$\m@th\mathord\leftarrow\mkern-6mu%
+  \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill
+  \mkern-6mu\mathord\rightarrow$}
+\def\overlrarrow#1{\vbox{\ialign{##\crcr
+      \lrarrowfill\crcr\noalign{\kern-\p@\nointerlineskip}
+      $\hfil\displaystyle{#1}\hfil$\crcr}}}
+
+\def\Line#1{\!\overlrarrow{\vphantom{t}\smash{\,#1\,}}\!}
+\def\Ray#1{\overrightarrow{\vphantom{t}#1\,}\!}
+\def\Seg#1{\overline{\vphantom{t}#1}}
+
+% ---- TEXT STYLES ----
+\def\mathsc#1{\text{\textsc{#1}}}
+\def\mathbs#1{\text{\boldmath\ensuremath{#1}}}
+\def\EMPH#1{\textbf{\emph{\boldmath #1}}}
+
+% ----------------------------------------------------------------------
+%  \begin{bigabstract}...\end{bigabstract}
+%  For use in documents with title pages.  Use normal-sized text in
+%  the abstract!
+% ----------------------------------------------------------------------
+\newenvironment{bigabstract}%
+{\begin{center}\large\textbf{\abstractname}\end{center}\begin{quotation}}%
+{\end{quotation}}
+
+% ----------------------------------------------------------------------
+%  Make captions smaller than the text, make their titles bold.
+%  Arguments:   #1 == figure name: "Figure 5"
+%               #2 == caption: "Papers by \Erdos, 1900--2000 (projected)"
+%  Less space after floats and before caption, since captions are smaller
+% ----------------------------------------------------------------------
+\@ifundefined{abovecaptionskip}{\newlength\abovecaptionskip}
+\long\def\@makecaption#1#2{
+   \vskip \abovecaptionskip
+   \setbox\@tempboxa\hbox{{\sffamily\footnotesize \textbf{#1.} #2}}
+   \ifdim \wd\@tempboxa >\hsize                     % if more than one line,
+       {\sffamily\footnotesize \textbf{#1.} #2\par}     %     typeset normally
+     \else                                          % else
+       \hbox to\hsize{\hfil\box\@tempboxa\hfil} %     center
+   \fi}
+\dbltextfloatsep 18pt plus 2pt minus 4pt    % was 20pt plus 2pt minus 4pt
+\textfloatsep 18pt plus 2pt minus 4pt       % was 20pt plus 2pt minus 4pt
+\abovecaptionskip 6pt                   % was 10pt
+
+% ----------------------------------------------------------------------
+%  Revised theorem environment:
+%  Add a period after the theorem number, and make theorems slanted
+%  instead of italic to help distinguish text from math.  Use
+%  \boldmath in theorem label in case it includes math.
+% ----------------------------------------------------------------------
+\def\@opargbegintheorem#1#2#3{\trivlist
+    \item[\hskip\labelsep{\bfseries\boldmath #1\ #2\ (#3).}]\slshape}
+\def\@begintheorem#1#2{\trivlist
+    \item[\hskip\labelsep{\bfseries\boldmath #1\ #2.}]\slshape}
+
+% ----------------------------------------------------------------------
+%  \newproof{type}{text}(style)[post]:
+%  Define a new type of unnumbered ``theorem'' environment.  The last
+%  two arguments are optional; most environments will not use them.
+%
+%  To change label style, put new style command in label:
+%       \newproof{comment}{\textsf{Comment}}
+% ----------------------------------------------------------------------
+\def\newproof#1#2{\@ifnextchar({\@snproof{#1}{#2}}{\@snproof{#1}{#2}()}}
+\def\@snproof#1#2(#3){\@ifnextchar[{\@xnproof{#1}{#2}{#3}}
+                                   {\@xnproof{#1}{#2}{#3}[]}}
+
+\def\@xnproof#1#2#3[#4]{%\expandafter\@ifdefinable\csname #1\endcsname
+{\global\@namedef{#1}{\@prf{#2}{#3}}\global\@namedef{end#1}{\@endprf{#4}}}}
+
+\def\@prf#1#2{\@ifnextchar[{\@xprf{#1}{#2}}{\@yprf{#1}{#2}}}
+\def\@xprf#1#2[#3]{\@yprf{#1\ (#3)}{#2}}
+\def\@yprf#1#2{\begin{trivlist}\item[\hskip\labelsep{\bfseries\boldmath #1:}]#2}
+
+\def\@endprf#1{#1\end{trivlist}}
+
+%
+%  Standard proof envrionment: last line has a halmos at the right margin.
+%  Insert \qed to override default halmos placement.
+%  Incompatible with amsprf package!
+%
+\newif\ifneedqed
+\global\needqedfalse
+\@ifundefined{square}{\let\square\Box}{}    % grumble latex2e 
+% \def\QED{\ensuremath{{\square}}}
+\def\markatright#1{\leavevmode\unskip\nobreak\quad\hspace*{\fill}{#1}}
+\def\qed{\ifneedqed\markatright{\ensuremath{{\square}}}\global\needqedfalse\else\null\fi}
+\newproof{proof}{Proof\global\needqedtrue}[\qed]
+\newproof{sketch}{Proof Sketch\global\needqedtrue}[\qed]
+
+%
+%  'Fake' proof environment, for use with wrapfig and similar packages
+%
+\newenvironment{rawproof}%
+   {\begin{trivlist}\item[\hskip\labelsep\textbf{Proof:}\global\needqedtrue]}%
+   {\qed\end{trivlist}}
+
+%
+%  If the proof ends with a displayed equation, use \aftermath just
+%  before \end{proof} to put the halmos in the ``right'' place.  
+%  Doesn't work worth a damn. 
+%
+\def\aftermath{\par\vspace{-\belowdisplayskip}\vspace{-\parskip}\vspace{-\baselineskip}}
+
+% ----------------------------------------------------------------------
+%  One-shot theoremish environment.  For named things like ``Zorn's
+%  Lemma'' or ``The Death Leap Principle'' or ``The Zone Theorem''.
+%  To get one-shot proofish environment, put \normalfont inside.
+%  New improved version stolen from Erik Demaine.
+% ----------------------------------------------------------------------
+\newenvironment{oneshot}[1]{\@begintheorem{#1}{\unskip}}{\@endtheorem}
+
+% Meh.
+\def\Comment#1{\textsf{\textsl{$\langle\!\langle$#1\/$\rangle\!\rangle$}}}
+
+\def\beginc@de{\noindent\begin{center}\begin{minipage}{1in}\tt
+        \begin{tabbing}~~~~\=~~~~\=~~~~\=~~~~\=~~~~\=~~~~\=~~~~\=\kill}
+\def\endc@de{\end{tabbing}\end{minipage}\end{center}}
+
+\newenvironment{code}{\beginc@de}{\endc@de}
+
+%
+%  These only work with <1 line of text.  Use sparingly!!
+%
+\def\textul#1{\underline{\smash{#1}\vphantom{.}}}
+\def\strike#1{\ensuremath{\overline{\text{\smash{#1}\vphantom{.}}}}}
+
+% ----------------------------------------------------------------------
+%  Include a file verbatim.  Searches the TEXINPUTS path for the file,
+%  even though that's probably not what you really want.  Stolen from
+%  ``verbatimfiles.sty'' by Chris Rowley and others.  [Don't use their 
+%  \vertbatimlisting macro; it has some bizarre side-effects!]
+% ----------------------------------------------------------------------
+\def\verbinput#1{
+\begingroup\@verbatim\frenchspacing\@vobeyspaces\input#1\endgroup
+}
+
+% ----------------------------------------------------------------------
+%  \now  -- Current time in h:mm AM/PM format
+%  \mdyy -- Today's date in m/d/yy format.  Forget Y2K; this is for humans!
+% ----------------------------------------------------------------------
+\newcount\timehh\timehh=\time
+\divide\timehh by 60
+\newcount\timemm\timemm=\time
+\count255=\timehh
+\multiply\count255 by -60
+\advance\timemm by \count255
+\newif\iftimePM
+\ifnum\timehh>11 \timePMtrue\else\timePMfalse\fi
+\ifnum\timehh<1 \advance\timehh by 12\fi
+\ifnum\timehh>12 \advance\timehh by -12\fi
+\def\now{\number\timehh:\ifnum\timemm<10 0\fi\number\timemm
+         \iftimePM pm\else am\fi}
+\newcount\mdYY\mdYY=\year
+\count255=\year
+\divide\count255 by 100
+\multiply\count255 by 100
+\advance\mdYY by -\count255
+\def\mdyy{\number\month/\number\day/\ifnum\mdYY<10 0\fi\number\mdYY}
+
+
+% ----------------------------------------------------------------------
+%  Notes to myself.  The margin flags are broken, thanks to an
+%  incompatibility with the geometry package.
+% ----------------------------------------------------------------------
+\def\n@te#1{\textsf{\boldmath \textbf{$\langle\!\langle$#1$\rangle\!\rangle$}}\leavevmode}
+\def\n@tew@rn{\GenericWarning{}{AUTHOR WARNING: Unresolved \protect\note}}
+
+\def\n@ten@te#1{\marginpar
+    [\hfill\llap{\textcolor{red}{{#1}$\!\Longrightarrow$}}]
+    {\rlap{\textcolor{red}{$\Longleftarrow\!${#1}}}}}
+
+% Warning: The textcomp and mathdesign packages are incompatible!
+%\def\n@tedingb@t{\@ifundefined{textmusicalnote}
+%           {$\maltese$}            % if you don't have textcomp
+%           {\textmusicalnote}}     % if you do have textcomp
+
+\def\n@tedingb@t{\textcircled{$\maltese$}}
+
+\def\note#1{\textcolor{red}{\n@tew@rn\n@te{\n@ten@te{\n@tedingb@t}#1}}}
+
+\def\sidenote#1{\marginpar{\tiny\textsf{#1}}}
+
+\setlength{\marginparsep}{0.1in}
+\setlength{\marginparwidth}{\paperwidth}
+\addtolength{\marginparwidth}{-\oddsidemargin}
+\addtolength{\marginparwidth}{-\textwidth}
+\addtolength{\marginparwidth}{-1.0\marginparsep} 
+\addtolength{\marginparwidth}{-1.0in}
+
+%----------------------------------------------------------------------
+%  TODO sets enclosed text in a gray rectangle with a red border
+%  with small sans-serif text.
+%----------------------------------------------------------------------
+\definecolor{TODOcolor}{cmyk}{0.05,0,0,0}
+\definecolor{TODOtxtcolor}{cmyk}{0,1,1,0}
+
+\newenvironment{TODO}
+   {\fboxrule1.5pt\begin{lrbox}{\@tempboxa}
+    \begin{minipage}{0.9\columnwidth}\small\sffamily\color{TODOtxtcolor}}
+   {\end{minipage}
+    \end{lrbox}
+    \medskip\centerline{\fcolorbox{TODOtxtcolor}{TODOcolor}{\usebox{\@tempboxa}}}
+    \sidenote{\LARGE\color{Red}$\bigstar\bigstar\bigstar$}\medskip}
+
+
+%----------------------------------------------------------------------
+%  Bibliography aliases, so I can use mnemonic citation keys and
+%  geom.bib at the same time.  \bibalias{foo}{bar} makes \cite{foo}
+%  act (almost) exactly like \cite{bar}.  I don’t actually use this.
+%----------------------------------------------------------------------
+\def\bibalias#1#2{%
+    \global\@namedef{b@#1}{%
+        {\@ifundefined{b@#2}{\textsf{?}}{}\csname b@#2\endcsname}}}
+
+% ----------------------------------------------------------------------
+%  Draft notice.  Put \DRAFT on the title page, in place of \date.
+% ----------------------------------------------------------------------
+\def\draftnotice{Preliminary draft --- \mdyy\ --- Not for distribution}
+
+\def\DRAFT{
+\date{\fbox{\textsf{\textbf{\draftnotice}}}}
+\def\@oddfoot{\footnotesize\hss\fbox{\textbf{\textsf{\draftnotice}}}\hss}
+\let\@evenfoot\@oddfoot
+}
+
+
+%----------------------------------------------------------------------
+% 'cramped' list style, stolen from Jeff Vitter.  Doesn't always work.
+%----------------------------------------------------------------------
+\def\cramped
+  {\parskip\@outerparskip\@topsep\parskip\@topsepadd2pt\itemsep0pt
+}
+
+%----------------------------------------------------------------------
+%  More footnote symbols, please.
+%----------------------------------------------------------------------
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or
+   \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger
+   \or \ddagger\ddagger \or \mathsection\mathsection \or
+   \mathparagraph\mathparagraph \else\@ctrerr\fi}}
+
+%----------------------------------------------------------------------
+%  TeXbook style quotations -- I should really use the right font, tho.
+%----------------------------------------------------------------------
+\newenvironment{rightquote}[1]
+{\flushright\begin{minipage}{#1\textwidth}
+\parskip0pt\footnotesize\sffamily\slshape}
+{\end{minipage}\par\bigskip}
+\def\quotee#1{\par\vspace{0.5ex}\begin{flushright}\unskip\textup{--- #1}\end{flushright}\unskip}
+
+%----------------------------------------------------------------------
+%  Creative Commons license
+%----------------------------------------------------------------------
+\def\CCLicense#1{\vfill\vbox to 0pt{\centering\tiny\sffamily \textcopyright\ Copyright \number\year\ Chao Xu.\\ This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (\url{http://creativecommons.org/licenses/by-nc-sa/4.0/}).\\  Free distribution is strongly encouraged; commercial distribution is expressly forbidden.\quad  See {#1} for the most recent revision.\vss}}
+
+

ref.bib

@@ -317,4 +317,34 @@ series = {SODA '95}
 	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}}
+@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}}
+
+@inproceedings{arora_towards_2013,
+	address = {Berkeley, CA, USA},
+	title = {Towards a {Better} {Approximation} for {Sparsest} {Cut}?},
+	isbn = {978-0-7695-5135-7},
+	url = {http://ieeexplore.ieee.org/document/6686163/},
+	doi = {10.1109/FOCS.2013.37},
+	language = {en},
+	urldate = {2025-05-09},
+	booktitle = {2013 {IEEE} 54th {Annual} {Symposium} on {Foundations} of {Computer} {Science}},
+	publisher = {IEEE},
+	author = {Arora, Sanjeev and Ge, Rong and Sinop, Ali Kemal},
+	month = oct,
+	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},
+}