commit c2eae7f62c25098744747edafdb24f357767d232 Author: Yu Cong Date: Thu Apr 10 16:41:11 2025 +0800 first commit diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..1d67259 --- /dev/null +++ b/.gitignore @@ -0,0 +1,308 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bbl-SAVE-ERROR +*.bcf +*.bcf-SAVE-ERROR +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync +*.rubbercache +rubber.cache + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hypdoc +*.hd + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. +# Uncomment the next line to have this generated file ignored. +#*Notes.bib \ No newline at end of file diff --git a/chao.sty b/chao.sty new file mode 100644 index 0000000..9023509 --- /dev/null +++ b/chao.sty @@ -0,0 +1,532 @@ +% ========================-*- LaTeX -*-========================= +% +% chao.sty -- modified from jeffe.sty +% jeffe.sty can be found at http://web.engr.illinois.edu/~jeffe/pubs/latex.html +% +% ========================================================== + + +\ProvidesPackage{chao} + + +% remarkable size +\newif\ifchao@remarkable +\DeclareOption{remarkable}{\chao@remarkabletrue} +\PassOptionsToPackage{hyphens}{url} +\PassOptionsToPackage{unicode}{hyperref} +\PassOptionsToPackage{naturalnames}{hyperref} +\ProcessOptions + + + + + +\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} + +% st mary fonts +\RequirePackage{stmaryrd} % comment out if unavailable +\SetSymbolFont{stmry}{bold}{U}{stmry}{m}{n} +\RequirePackage{wasysym} +% nagging +\RequirePackage[l2tabu, orthodox]{nag} +\RequirePackage[all,warning]{onlyamsmath} + + + +\RequirePackage{graphicx} % essentials +\RequirePackage{microtype} +\RequirePackage[mathletters]{ucs} % allow Unicode in .tex file +\RequirePackage[utf8]{inputenc} +\RequirePackage[charter]{mathdesign} % change fonts +\RequirePackage{XCharter} % change fonts +\RequirePackage{berasans, beramono} +\RequirePackage{eucal} +\RequirePackage[nocompress]{cite} % other convenient stuff +\RequirePackage{enumerate} +%\usepackage{pgf,tikz} +\RequirePackage{wrapfig} +\RequirePackage[medium,compact]{titlesec} +\RequirePackage{verbatim} +\usepackage{CJKutf8} + + +\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 + +% line numbers (be nice to the referee) +\RequirePackage[mathlines, pagewise]{lineno} +%\linenumbers +\nolinenumbers +\renewcommand\linenumberfont{\normalfont\tiny\sffamily\color{gray}} +\setlength\linenumbersep{2em} + +\numberwithin{figure}{section} + +\ifchao@remarkable + \RequirePackage[paperheight=209.6mm, paperwidth=157.2mm, top=0.2in,bottom=0.2in,right=0.2in,left=0.2in]{geometry} + \tolerance=10000 +\else + \RequirePackage[margin=1in]{geometry} +\fi + +%\setlength{\topmargin}{-0.5in} \setlength{\textwidth}{6.5in} +%\setlength{\oddsidemargin}{0.0in} \setlength{\textheight}{9.1in} + +%\pagestyle{empty} +%\newlength{\pagewidth} +%\setlength{\pagewidth}{6.5in} %\pagestyle{empty} +\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 + + +% Chao addition +\def\email#1{\href{mailto:#1}{#1}} +% chinese +\def\zh#1{\begin{CJK}{UTF8}{gbsn}#1\end{CJK}} + + + +% ---------------------------------------------------------------------- +% 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\varepsilon % 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{\QED}\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}} + + diff --git a/main.pdf b/main.pdf new file mode 100644 index 0000000..e78ca71 Binary files /dev/null and b/main.pdf differ diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..2a65e27 --- /dev/null +++ b/main.tex @@ -0,0 +1,157 @@ +\documentclass[12pt]{article} +\usepackage{chao} +\title{connectivity interdiction} +\author{} +\date{} + +\DeclareMathOperator*{\opt}{OPT} +\DeclareMathOperator*{\len}{len} + +\begin{document} + + +\section{``Cut-free'' Proof} +\begin{problem}[b-free knapsack]\label{bfreeknap} + Consider a set of elements $E$ and two weights $w:E\to \Z_+$ and + $c:E\to Z_+$ and a budget $b\in \Z_+$. Given a feasible set $\mathcal + F\subset 2^E$, find $\min_{X + \in \mathcal F, F\subset E} w(X\setminus F)$ such that $c(F)\leq b$. +\end{problem} +Always remember that $\mathcal F$ is usually not explicitly given. + +\begin{problem}[Normalized knapsack]\label{nknap} + Given the same input as \autoref{bfreeknap}, find $\min \limits_{X + \in \mathcal F, F\subset E} \frac{w(X\setminus F)}{b+1-c(F)}$ such that + $c(F)\leq b$. +\end{problem} + +Denote by $\tau$ the optimum of \autoref{nknap}. Define a new weight +$w_\tau:\E\to \R$, + +\[ +w_\tau(e)=\begin{cases} + w(e) & \text{if $w(e)< \tau\cdot c(e)$ (light elem)}\\ + \tau\cdot c(e) & \text{otherwise (heavy elem)} +\end{cases} +\] + +\begin{lemma} + Let $(X^N,F^N)$ be the optimal solution to \autoref{nknap}. + Every element in $F^N$ is heavy. +\end{lemma} +proof is exactly the same as lemma 1 in \cite{vygen_fptas_2024}. + +\begin{lemma}\label{lem:lb} + For any $X\in \mathcal F$, $w_\tau(X)\ge \tau(1+b)$. +\end{lemma} +proof is the same. + +\begin{lemma} + % $w_\tau (X^N)\le \tau(b+1)$. + $X^N\in \arg\min\limits_{X\in\mathcal F} w_\tau(X)$. +\end{lemma} +\begin{proof} + \begin{align*} + w_\tau (X^N) & \le w(X^N\setminus F^N) + w_\tau(F^N)\\ + & = \tau \cdot(b+1-c(F^N)) + \tau\cdot c(F^N)\\ + & = \tau(b+1) + \end{align*} + Thus by \autoref{lem:lb}, $X^N$ gets the minimum. +\end{proof} + +Let $(X^*,F^*)$ be the optimal solution to \autoref{bfreeknap}. + +\begin{lemma} + $X^*$ is either an $\alpha$-approximate solution to $\min_{X\in\mathcal F} + w_\tau(X)$ for some $\alpha>1$, or $w(X^*\setminus F^*)\geq + \tau(\alpha+(\alpha-1)b)$. +\end{lemma} +The proof is the same. + +In fact, corollary 1 and theorem 5 are also the same as those in +\cite{vygen_fptas_2024}. Finally we get a knapsack version of Theorem 4: +\begin{theorem}[Theorem 4 in \cite{vygen_fptas_2024}] + Let $X^{\min}$ be the optimal solution to $\min_{X\in\mathcal F} w_\tau(X)$. + The optimal set $X^*$ in \autoref{bfreeknap} is a + 2-approximation to $X^{\min}$. +\end{theorem} + +Thus to obtain a FPTAS for \autoref{bfreeknap}, one need to design a FPTAS for +\autoref{nknap} and a polynomial time alg for finding all 2-approximations to +$\min_{X\in\mathcal F} w_\tau(X)$. + +\paragraph{FPTAS for \autoref{nknap} in \cite{vygen_fptas_2024}} (The name +``FPTAS'' here is not precise since we do not have a approximation scheme but +an enumeration algorithm. But I will use this term anyway.) In their settings, +$\mathcal F$ is the collection of all cuts in some graph. +Let $\opt^N$ be the optimum of \autoref{nknap}. We can assume that there is no +$X\in \mathcal F$ s.t. $c(X)\le b$ since this is polynomially detectable +(through min-cut on $c(\cdot)$) and the optimum is 0. Thus we have +$\frac{1}{b+1} \le \opt^N \le |E|\cdot \max_e w(e)$. Then we enumerate +$\frac{(1+\varepsilon)^i}{b+1}$ where $i\in \set{0,1,\ldots,\floor{\log_{1+ +\varepsilon}(|E|w_{\max}(b+1))}}$. There is a feasible $i$ s.t. $(1-\varepsilon) +\opt^N\le \frac{(1+\varepsilon)^i}{b+1}\leq \opt^N$ since +$\frac{(1+\varepsilon)^i}{b+1}\le \opt^N\le \frac{(1+\varepsilon)^{i+1}}{b+1}$ +holds for some $i$. + +Note that this enumeration scheme also holds for arbitrary $\mathcal F$ if we +have a non-zero lowerbound on $\opt^N$. + +\begin{conjecture} +Let $(C,F)$ be the optimal solution to connectivity interdiction. The optimum +cut $C$ can be computed in polynomial time. In other words, connectivity +interdiction is almost as easy as knapsack. +\end{conjecture} + +\section{Connections} +For unit costs, connectivity interdiction with budget $b=k-1$ is the same +problem as finding the minimum weighted edge set whose removal breaks $k$-edge +connectivity. + +It turns out that \autoref{nknap} is just a necessary ingredient for MWU. +Authors of \cite{vygen_fptas_2024} $\subset$ authors of +\cite{chalermsook_approximating_2022}. + +How to derive normalized min cut for connectivity interdiction? + + + +\section{Random Stuff} + +\subsection{remove box constraints} +Given a positive covering LP, + +\begin{equation*} +\begin{aligned} +LP1=\min& & \sum_e w(e) x_e& & & \\ +s.t.& & \sum_{e\in T} c(e)x_e&\geq k & &\forall T\\ +& & c(e)\geq x_e&\geq 0 & &\forall e, +\end{aligned} +\end{equation*} +we want to remove constraints $c(e)\geq x_e$. Consider the following LP, +\begin{equation*} +\begin{aligned} +LP2=\min& & \sum_e w(e) x_e& & & \\ +s.t.& & \sum_{e\in T} c(e)x_e&\geq k & &\forall T\\ +& & \sum_{e\in T\setminus f} c(e)x_e&\geq k - c(f) & &\forall T \; + \forall f\in T\\ +& & x_e&\geq 0 & &\forall e, +\end{aligned} +\end{equation*} + +These two LPs have the same optimum. One can see that any feasible solution to +LP1 is feasible in LP2. Thus $\opt(LP1) \geq \opt(LP2)$. Next we show that any +$x_e$ in the optimum solution to LP2 is always in $[0,c(e)]$. +Let $x^*$ be the optimum and +suppose that $c(f)k$ since we assume $x_f>c(f)$, which means we can decrease $x_f$ without violating any constraint. Thus it contradicts the assumption that $x^*$ is optimal. Then we can add redundant constraints $x_e\leq c(e) \;\forall e$ to LP2 and see that LP1 and LP2 have the same optimum. + +This applies to \cite{chalermsook_approximating_2022} but cannot get an improvement on their algorithm.(MWU does not care the number of constraints.) So does this trick apply to connectivity interdiction? + +\[ +\min_{\text{cut C}, f\in C}\frac{\sum_{e\in C\setminus\set{f}}w(e)x_e}{k-c(f)} +\] +\bibliographystyle{plain} +\bibliography{ref} +\end{document} diff --git a/ref.bib b/ref.bib new file mode 100644 index 0000000..414f8d6 --- /dev/null +++ b/ref.bib @@ -0,0 +1,59 @@ + +@inproceedings{vygen_fptas_2024, + address = {Cham}, + title = {An {FPTAS} for {Connectivity} {Interdiction}}, + volume = {14679}, + isbn = {978-3-031-59834-0 978-3-031-59835-7}, + url = {https://link.springer.com/10.1007/978-3-031-59835-7_16}, + language = {en}, + urldate = {2024-11-04}, + booktitle = {Integer {Programming} and {Combinatorial} {Optimization}}, + publisher = {Springer Nature Switzerland}, + author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai}, + editor = {Vygen, Jens and Byrka, Jarosław}, + year = {2024}, + doi = {10.1007/978-3-031-59835-7_16}, + pages = {210--223}, +} + +@article{chalermsook_approximating_2022, + title = {Approximating k-{Edge}-{Connected} {Spanning} {Subgraphs} via a {Near}-{Linear} {Time} {LP} {Solver}}, + volume = {229}, + copyright = {Creative Commons Attribution 4.0 International license, info:eu-repo/semantics/openAccess}, + issn = {1868-8969}, + url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.37}, + doi = {10.4230/LIPICS.ICALP.2022.37}, + language = {en}, + urldate = {2025-03-09}, + journal = {LIPIcs, Volume 229, ICALP 2022}, + author = {Chalermsook, Parinya and Huang, Chien-Chung and Nanongkai, Danupon and Saranurak, Thatchaphol and Sukprasert, Pattara and Yingchareonthawornchai, Sorrachai}, + editor = {Bojańczyk, Mikołaj and Merelli, Emanuela and Woodruff, David P.}, + year = {2022}, + keywords = {Approximation Algorithms, Data Structures, Theory of computation → Routing and network design problems}, + pages = {37:1--37:20}, +} + + + + + + + + +@article{garg_faster_nodate, +author = {Garg, Naveen and K\"{o}nemann, Jochen}, +title = {Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems}, +journal = {SIAM Journal on Computing}, +volume = {37}, +number = {2}, +pages = {630-652}, +year = {2007}, +doi = {10.1137/S0097539704446232}, +URL = { https://doi.org/10.1137/S0097539704446232}, +eprint = {https://doi.org/10.1137/S0097539704446232}, +abstract = { This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems. } +} + + + +