hw12-matching-flow-solution: +template

courses-at-nju-by-hfwei · Jun 14, 2021 · ce7ca42 · ce7ca42
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diff --git a/2021/solution/hw12-matching-flow-solution/README.md b/2021/solution/hw12-matching-flow-solution/README.md
@@ -0,0 +1 @@
+# README.md
diff --git a/2021/solution/hw12-matching-flow-solution/hw-preamble.tex b/2021/solution/hw12-matching-flow-solution/hw-preamble.tex
@@ -0,0 +1,198 @@
+% hw-preamble.tex
+
+% geometry for A4 paper
+% See https://tex.stackexchange.com/a/119912/23098
+\geometry{
+  left=20.0mm,
+  top=20.0mm,
+  bottom=20.0mm,
+  textwidth=130mm, % main text block
+  marginparsep=5.0mm, % gutter between main text block and margin notes
+  marginparwidth=50.0mm % width of margin notes
+}
+
+% for colors
+\usepackage{xcolor} % usage: \color{red}{text}
+% predefined colors
+\newcommand{\red}[1]{\textcolor{red}{#1}} % usage: \red{text}
+\newcommand{\blue}[1]{\textcolor{blue}{#1}}
+\newcommand{\teal}[1]{\textcolor{teal}{#1}}
+
+\usepackage{todonotes}
+
+% heading
+\usepackage{sectsty}
+\setcounter{secnumdepth}{2}
+\allsectionsfont{\centering\huge\rmfamily}
+
+% for Chinese
+\usepackage{xeCJK}
+\usepackage{zhnumber}
+\setCJKmainfont[BoldFont=FandolSong-Bold.otf]{FandolSong-Regular.otf}
+
+% for fonts
+\usepackage{fontspec}
+\newcommand{\song}{\CJKfamily{song}}
+\newcommand{\kai}{\CJKfamily{kai}}
+
+% To fix the ``MakeTextLowerCase'' bug:
+% See https://github.com/Tufte-LaTeX/tufte-latex/issues/64#issuecomment-78572017
+% Set up the spacing using fontspec features
+\renewcommand\allcapsspacing[1]{{\addfontfeature{LetterSpace=15}#1}}
+\renewcommand\smallcapsspacing[1]{{\addfontfeature{LetterSpace=10}#1}}
+
+% for url
+\usepackage{hyperref}
+\hypersetup{colorlinks = true,
+  linkcolor = teal,
+  urlcolor  = teal,
+  citecolor = blue,
+  anchorcolor = blue}
+
+\newcommand{\me}[4]{
+    \author{
+      {\bfseries 姓名：}\underline{#1}\hspace{2em}
+      {\bfseries 学号：}\underline{#2}\hspace{2em}\\[10pt]
+      {\bfseries 评分：}\underline{#3\hspace{3em}}\hspace{2em}
+      {\bfseries 评阅：}\underline{#4\hspace{3em}}
+  }
+}
+
+% Please ALWAYS Keep This.
+\newcommand{\noplagiarism}{
+  \begin{center}
+    \fbox{\begin{tabular}{@{}c@{}}
+      请独立完成作业，不得抄袭。\\
+      若得到他人帮助, 请致谢。\\
+      若参考了其它资料，请给出引用。\\
+      鼓励讨论，但需独立书写解题过程。
+    \end{tabular}}
+  \end{center}
+}
+
+% \newcommand{\goal}[1]{
+%   \begin{center}{\fcolorbox{blue}{yellow!60}{\parbox{0.50\textwidth}{\large
+%     \begin{itemize}
+%       \item 体会``思维的乐趣''
+%       \item 初步了解递归与数学归纳法
+%       \item 初步接触算法概念与问题下界概念
+%     \end{itemize}}}}
+%   \end{center}
+% }
+
+% Each hw consists of four parts:
+\newcommand{\beginrequired}{\hspace{5em}\section{作业 (必做部分)}}
+\newcommand{\beginoptional}{\section{作业 (选做部分)}}
+\newcommand{\beginot}{\section{Open Topics}}
+\newcommand{\begincorrection}{\section{订正}}
+\newcommand{\beginfb}{\section{反馈}}
+
+% for math
+\usepackage{amsmath, mathtools, amsfonts, amssymb}
+\newcommand{\set}[1]{\{#1\}}
+\newcommand{\ps}[1]{\mathcal{P}(#1)}
+
+% define theorem-like environments
+\usepackage[amsmath, thmmarks]{ntheorem}
+
+\theoremstyle{break}
+\theorempreskip{2.0\topsep}
+\theorembodyfont{\song}
+\theoremseparator{}
+\newtheorem{problem}{题目}[subsection]
+\renewcommand{\theproblem}{\arabic{problem}}
+\newtheorem{definition}{定义}[subsection]
+\renewcommand{\thedefinition}{\arabic{definition}}
+\newtheorem{theorem}{定理}[subsection]
+\renewcommand{\thetheorem}{\arabic{theorem}}
+\newtheorem{ot}{Open Topics}
+
+\theorempreskip{3.0\topsep}
+\theoremheaderfont{\kai\bfseries}
+\theoremseparator{：}
+\theorempostwork{\bigskip\hrule}
+\newtheorem*{solution}{解答}
+\theorempostwork{\bigskip\hrule}
+\newtheorem*{revision}{订正}
+
+\theoremstyle{plain}
+\newtheorem*{cause}{错因分析}
+\newtheorem*{remark}{注}
+
+\theoremstyle{break}
+\theorempostwork{\bigskip\hrule}
+\theoremsymbol{\ensuremath{\Box}}
+\newtheorem*{proof}{证明}
+
+% \newcommand{\ot}{\blue{\bf [OT]}}
+
+% for figs
+\renewcommand\figurename{图}
+\renewcommand\tablename{表}
+
+% for fig without caption: #1: width/size; #2: fig file
+\newcommand{\fig}[2]{
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[#1]{#2}
+  \end{figure}
+}
+% for fig with caption: #1: width/size; #2: fig file; #3: caption
+\newcommand{\figcap}[3]{
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[#1]{#2}
+    \caption{#3}
+  \end{figure}
+}
+% for fig with both caption and label: #1: width/size; #2: fig file; #3: caption; #4: label
+\newcommand{\figcaplbl}[4]{
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[#1]{#2}
+    \caption{#3}
+    \label{#4}
+  \end{figure}
+}
+% for margin fig without caption: #1: width/size; #2: fig file
+\newcommand{\mfig}[2]{
+  \begin{marginfigure}
+    \centering
+    \includegraphics[#1]{#2}
+  \end{marginfigure}
+}
+% for margin fig with caption: #1: width/size; #2: fig file; #3: caption
+\newcommand{\mfigcap}[3]{
+  \begin{marginfigure}
+    \centering
+    \includegraphics[#1]{#2}
+    \caption{#3}
+  \end{marginfigure}
+}
+
+\usepackage{fancyvrb}
+
+% for algorithms
+\usepackage[]{algorithm}
+\usepackage[]{algpseudocode} % noend
+% See [Adjust the indentation whithin the algorithmicx-package when a line is broken](https://tex.stackexchange.com/a/68540/23098)
+\newcommand{\algparbox}[1]{\parbox[t]{\dimexpr\linewidth-\algorithmicindent}{#1\strut}}
+\newcommand{\hStatex}[0]{\vspace{5pt}}
+\makeatletter
+\newlength{\trianglerightwidth}
+\settowidth{\trianglerightwidth}{$\triangleright$~}
+\algnewcommand{\LineComment}[1]{\Statex \hskip\ALG@thistlm \(\triangleright\) #1}
+\algnewcommand{\LineCommentCont}[1]{\Statex \hskip\ALG@thistlm%
+  \parbox[t]{\dimexpr\linewidth-\ALG@thistlm}{\hangindent=\trianglerightwidth \hangafter=1 \strut$\triangleright$ #1\strut}}
+\makeatother
+
+% for footnote/marginnote
+% see https://tex.stackexchange.com/a/133265/23098
+\usepackage{tikz}
+\newcommand{\circled}[1]{%
+  \tikz[baseline=(char.base)]
+  \node [draw, circle, inner sep = 0.5pt, font = \tiny, minimum size = 8pt] (char) {#1};
+}
+\renewcommand\thefootnote{\protect\circled{\arabic{footnote}}}
+
+\newcommand{\score}[1]{{\bf [#1 分]}}
diff --git a/2021/solution/hw12-matching-flow-solution/hw12-matching-flow-solution.pdf b/2021/solution/hw12-matching-flow-solution/hw12-matching-flow-solution.pdf
diff --git a/2021/solution/hw12-matching-flow-solution/hw12-matching-flow-solution.tex b/2021/solution/hw12-matching-flow-solution/hw12-matching-flow-solution.tex
@@ -0,0 +1,104 @@
+% hw12-matching-flow-solution.tex
+
+% !TEX program = xelatex
+%%%%%%%%%%%%%%%%%%%%
+% see http://mirrors.concertpass.com/tex-archive/macros/latex/contrib/tufte-latex/sample-handout.pdf
+% for how to use tufte-handout
+\documentclass[a4paper, justified]{tufte-handout}
+
+\input{hw-preamble} % feel free to modify this file if you understand LaTeX well
+%%%%%%%%%%%%%%%%%%%%
+\title{12. 图论: 匹配与网络流 (12-matching-flow)}
+\me{魏恒峰}{hfwei@nju.edu.cn}{}{}
+\date{2021年05月28日 发布作业 \\ 2021年06月xx日 发布答案}
+%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+\maketitle
+%%%%%%%%%%%%%%%%%%%%
+\noplagiarism % PLEASE DON'T DELETE THIS LINE!
+%%%%%%%%%%%%%%%%%%%%
+\begin{abstract}
+  % \mfigcap{width = 0.85\textwidth}{figs/George-Boole}{George Boole}
+  % \begin{center}{\fcolorbox{blue}{yellow!60}{\parbox{0.65\textwidth}{\large
+  %   \begin{itemize}
+  %     \item
+  %   \end{itemize}}}}
+  % \end{center}
+\end{abstract}
+%%%%%%%%%%%%%%%%%%%%
+\beginrequired
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{5 = 2 + 3} $\star\star$]
+  设 $G = (X, Y, E)$ 是一个 $k$-正则 ($k > 0$) 二部图。
+  请证明:
+  \begin{enumerate}[(1)]
+    \item $|X| = |Y|$;
+    \item $G$ 有一个 $X$-完美匹配。
+  \end{enumerate}
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{5} $\star\star\star$]
+  设 $G = (V, E)$ 是含有 $2n$ 个顶点的简单图，且 $\delta(G) \ge n + 1$。\\
+  请证明: $G$ 有完美匹配~\footnote{对于任意图, 完美匹配是 cover 了所有顶点的匹配。}。
+  (提示: 考虑使用图论第一讲中的定理。)
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{5} $\star\star\star\star$]
+  请证明: 每个二部图 $G$ 都有一个大小 $\ge e(G)/\Delta(G)$ 的匹配~\footnote{$e(G)$表示$G$的边数。}。
+  (提示: 使用 K\"{o}nig-Egerv\'{a}ry 定理。)
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%
+\begin{problem}[\score{5} $\star\star$]
+  设 $Y$ 为集合,
+  $\mathcal{A} = \set{A_{1}, \dots, A_{m}}$
+  为包含 $m$ 个集合的集合, 其中 $A_{i} \subseteq Y$ (对 $1 \le i \le m$)。
+  $\mathcal{A}$ 的相异代表系 (System of Distinct Representatives; SDR)
+  是 $Y$ 中 $m$ 个不同元素 $a_{1}, \dots, a_{m}$ 构成的集合,
+  其中 $a_{i} \in A_{i}$ (对 $1 \le i \le m$)。
+
+  \noindent 请证明: $\mathcal{A}$ 有 SDR 当且仅当
+  \[
+    \forall S \subseteq \set{1, \dots, m}.\;
+      \big\lvert \bigcup_{i \in S} A_{i} \big\rvert \ge |S|.
+  \]
+\end{problem}
+
+\begin{proof}
+\end{proof}
+%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%
+% 如果没有需要订正的题目，可以把这部分删掉
+\begincorrection
+%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%
+% 如果没有反馈，可以把这部分删掉
+\beginfb
+
+你可以写 (也可以发邮件或者使用``教学立方'')
+\begin{itemize}
+  \item 对课程及教师的建议与意见
+  \item 教材中不理解的内容
+  \item 希望深入了解的内容
+  \item $\cdots$
+\end{itemize}
+%%%%%%%%%%%%%%%%%%%%
+\end{document}
diff --git a/2021/solution/hw12-matching-flow-solution/tufte-book.cls b/2021/solution/hw12-matching-flow-solution/tufte-book.cls
@@ -0,0 +1,80 @@
+\NeedsTeXFormat{LaTeX2e}[1994/06/01]
+
+\ProvidesClass{tufte-book}[2009/12/11 v3.5.0 Tufte-book class]
+
+%%
+% Declare we're tufte-book
+\newcommand{\@tufte@class}{book}% the base LaTeX class (defaults to the article/handout style)
+\newcommand{\@tufte@pkgname}{tufte-book}% the name of the package (defaults to tufte-handout)
+
+%%
+% Load the common style elements
+\input{tufte-common.def}
+
+
+%%
+% Set up any book-specific stuff now
+
+%%
+% The front matter in Tufte's /Beautiful Evidence/ contains everything up
+% to the opening page of Chapter 1.  The running heads, when they appear,
+% contain only the (arabic) page number in the outside corner.
+%\newif\if@mainmatter \@mainmattertrue
+\renewcommand\frontmatter{%
+  \cleardoublepage%
+  \@mainmatterfalse%
+  \pagenumbering{arabic}%
+  %\pagestyle{plain}%
+  \fancyhf{}%
+  \ifthenelse{\boolean{@tufte@twoside}}%
+    {\fancyhead[LE,RO]{\thepage}}%
+    {\fancyhead[RE,RO]{\thepage}}%
+}
+
+
+%%
+% The main matter in Tufte's /Beautiful Evidence/ doesn't restart the page
+% numbering---it continues where it left off in the front matter.
+\renewcommand\mainmatter{%
+  \cleardoublepage%
+  \@mainmattertrue%
+  \fancyhf{}%
+  \ifthenelse{\boolean{@tufte@twoside}}%
+    {% two-side
+      \renewcommand{\chaptermark}[1]{\markboth{##1}{}}%
+      \fancyhead[LE]{\thepage\quad\smallcaps{\newlinetospace{\plaintitle}}}% book title
+      \fancyhead[RO]{\smallcaps{\newlinetospace{\leftmark}}\quad\thepage}% chapter title
+    }%
+    {% one-side
+      \fancyhead[RE,RO]{\smallcaps{\newlinetospace{\plaintitle}}\quad\thepage}% book title
+    }%
+}
+
+
+%%
+% The back matter contains appendices, indices, glossaries, endnotes,
+% biliographies, list of contributors, illustration credits, etc.
+\renewcommand\backmatter{%
+  \if@openright%
+    \cleardoublepage%
+  \else%
+    \clearpage%
+  \fi%
+  \@mainmatterfalse%
+}
+
+%%
+% Only show the chapter titles in the table of contents
+\setcounter{tocdepth}{0}
+
+%%
+% If there is a `tufte-book-local.sty' file, load it.
+
+\IfFileExists{tufte-book-local.tex}
+  {\input{tufte-book-local}
+   \TufteInfoNL{Loading tufte-book-local.tex}}
+  {}
+
+%%
+% End of file
+\endinput