first commit
This commit is contained in:
commit
c2eae7f62c
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.gitignore
vendored
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308
.gitignore
vendored
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@ -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
|
||||
532
chao.sty
Normal file
532
chao.sty
Normal file
@ -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}}
|
||||
|
||||
|
||||
157
main.tex
Normal file
157
main.tex
Normal file
@ -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 $c(f)<x_f\in x^*$. Consider all constraints $\sum_{e\in T\setminus f} c(e)
|
||||
x_e\geq k-c(f)$ on $T\ni f$. For any such constraint, we have $\sum_{e\in T} c(e)
|
||||
x_e>k$ 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}
|
||||
59
ref.bib
Normal file
59
ref.bib
Normal file
@ -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. }
|
||||
}
|
||||
|
||||
|
||||
|
||||
|
||||
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Reference in New Issue
Block a user