From f82e044c0a3664371694d860cffbdc4b6786cec7 Mon Sep 17 00:00:00 2001 From: Armin Friedl Date: Thu, 30 Jun 2016 01:36:46 +0200 Subject: [PATCH] finished --- _region_.pdf | 0 _region_.tex | 284 +++++++++++++++++++++++++++++++++++++++++ inputs/elimination.tex | 8 +- inputs/results.tex | 20 +++ inputs/separators.tex | 80 ++++++++++++ res/results.png | Bin 0 -> 205850 bytes res/widetree.png | Bin 0 -> 204150 bytes upperbounds.pdf | Bin 68843 -> 227332 bytes upperbounds.tex | 44 +++++++ 9 files changed, 432 insertions(+), 4 deletions(-) create mode 100644 _region_.pdf create mode 100644 _region_.tex create mode 100644 inputs/results.tex create mode 100644 res/results.png create mode 100644 res/widetree.png diff --git a/_region_.pdf b/_region_.pdf new file mode 100644 index 0000000..e69de29 diff --git a/_region_.tex b/_region_.tex new file mode 100644 index 0000000..b1ec086 --- /dev/null +++ b/_region_.tex @@ -0,0 +1,284 @@ +\message{ !name(../upperbounds.tex)}\documentclass[10pt,dvipsnames]{beamer} +\usepackage{appendixnumberbeamer} +\usepackage{booktabs} +\usepackage[scale=2]{ccicons} +\usepackage{pgfplots} +\usepgfplotslibrary{dateplot} +\usepackage{xspace} +%\usepackage{graphicx} +\usepackage{scalerel} +\usepackage[normalem]{ulem} +\usepackage[noend]{algorithm2e} + +% == General Beamer Settings == +\usetheme{metropolis} +\setbeamertemplate{blocks}[rounded] +\metroset{block=fill} +\usefonttheme{professionalfonts} +\usepackage{mathspec} +\setsansfont{Fira Sans Light} +\setmathsfont{Oldstyle} + + +% == Presentation Settings == +\def\thumbsup{\scalerel*{\includegraphics{res/up.png}}{O}} +\def\thumbsdown{\scalerel*{\includegraphics{res/down.png}}{O}} + +\subject{Upper Bounds} +\title[Upper Bounds]{Treewidth computations I. Upper bounds} +\subtitle{Hans L. Bodlaender, Arie M.C.A Koster} +\author{Armin Friedl} +\date{\today} + +\begin{document} + +\message{ !name(inputs/elimination.tex) !offset(-35) } +\begin{frame}{Table of contents} + \setbeamertemplate{section in toc}[sections numbered] + \setbeamertemplate{subsection in toc}[square] + \tableofcontents[sections={2}] +\end{frame} + + +\subsection{Idea} + +\begin{frame}{Idea} + \setbeamercolor{block title}{fg=RoyalBlue!70} + \begin{block}{Theorem {\normalfont \small \color{black} \cite{bodlaender2010, gavril1974}}} + Equivalent: + \begin{enumerate}[(i)] + \item $G$ has a treewidth at most k. + \item There is an elimination ordering $\pi$, such that no vertex $v\in V$ has more than $k$ neighbours with a higher number in $\pi$ in $G^+_\pi$ + \end{enumerate} + \end{block} + + \setbeamercolor{block title}{fg=ForestGreen!70} + \begin{block}{Application} + \begin{enumerate} + \item Take \emph{some} elimination ordering $\pi$ of $G$ + \item Construct $G^+_\pi$, calculate $k$ + \item $\xrightarrow{(i)~\equiv~(ii)}$ Upper Bound for treewidth + \end{enumerate} + \end{block} +\end{frame} + +\begin{frame}{What is $\bf G^+_\pi$ ?} + \metroset{block=transparent} + \begin{columns}[c] + \begin{column}{0.5\textwidth} + + \begin{block}{}\centering + \begin{tikzpicture} + \node[shape=circle,draw=black] (A) at (0,0) {A}; + \only<2>{\node[shape=circle,fill=mLightGreen] (A) at (0,0) {A};} + \only<3->{\node[shape=circle,fill=mDarkTeal,text=white] (A) at (0,0) {A};} + + \node[shape=circle,draw=black] (B) at (3,0) {B}; + \only<2>{\node[shape=circle,fill=mLightBrown] (B) at (3,0) {B};} + \only<3>{\node[shape=circle,fill=mLightGreen] (B) at (3,0) {B};} + \only<4->{\node[shape=circle,fill=mDarkTeal,text=white] (B) at (3,0) {B};} + + \node[shape=circle,draw=black] (C) at (0,3) {C}; + \only<2>{\node[shape=circle,fill=mLightBrown] (C) at (0,3) {C};} + \only<3>{\node[shape=circle,fill=mLightBrown] (C) at (0,3) {C};} + \only<4>{\node[shape=circle,fill=mLightGreen] (C) at (0,3) {C};} + \only<5->{\node[shape=circle,fill=mDarkTeal,text=white] (C) at (0,3) {C};} + + \node[shape=circle,draw=black] (D) at (3,3) {D}; + \only<3>{\node[shape=circle,fill=mLightBrown] (D) at (3,3) {D};} + \only<4>{\node[shape=circle,fill=mLightBrown] (D) at (3,3) {D};} + \only<5>{\node[shape=circle,fill=mLightGreen] (D) at (3,3) {D};} + \only<6->{\node[shape=circle,fill=mDarkTeal,text=white] (D) at (3,3) {D};} + + \node[shape=circle,draw=black] (E) at (1.5,5) {E}; + \only<4,5>{\node[shape=circle,fill=mLightBrown] (E) at (1.5,5) {E};} + \only<6>{\node[shape=circle,fill=mLightGreen] (E) at (1.5,5) {E};} + \only<7>{\node[shape=circle,fill=mDarkTeal,text=white] (E) at (1.5,5) {E};} + + \path (A) edge (B); + \path (A) edge (C); + \path (B) edge (D); + \path (C) edge (D); + \path (C) edge (E); + + + \path[style=dashed, color=mLightBrown]<2> (B) edge (C); + \only<3->{\path[style=dashed, color=mDarkTeal] (B) edge (C);} + + \path[style=dashed, color=mLightBrown]<4> (E) edge (D); + \only<5->{\path[style=dashed, color=mDarkTeal] (E) edge (D);} + \end{tikzpicture} + \end{block}{} + + \begin{block}{}\centering + $\pi = [\textcolor<2>{mLightGreen}{A} + \textcolor<3>{mLightGreen}{,B} + \textcolor<4>{mLightGreen}{,C} + \textcolor<5>{mLightGreen}{,D} + \textcolor<6>{mLightGreen}{,E}]$ + \end{block}{} + \end{column} + + \begin{column}{0.5\textwidth} + \only<1-6>{ + \begin{algorithm}[H] + \label{alg:fill} + \KwIn{$G, \pi$} + \KwOut{$G^+_{\pi}$} + $H=G$\\ + \ForEach{$v\in V_G$}{ + \ForEach{$w, x$ of N$_H$($v$)}{ + \If{$\pi(w), \pi(x)>\pi(v)$}{ + \alert<2->{add \{w,x\} to E$_H$} + } + } + } + \KwRet{H} + \end{algorithm} + } + + \only<7>{ + \begin{itemize} + \item $G^+_\pi$ is chordal + \item $G$ is a subgraph of $G^+_\pi$ + \item $\pi$ is a perfect elimination ordering of $G^+_\pi$ + \item $tw$ of \emph{subtree graph} (also a tree decomposition) of $G^+_\pi$ is $\text{MAXCLIQUE}(G^+_\pi)-1$ ~\cite{gavril1974} + \item There is a tree decomposition algorithm for $G$ with $tw = \text{MAXCLIQUE}(G^+_\pi)-1$, polynomial in n ~\cite{bodlaender2010} + \end{itemize} + } + + \end{column} +\end{columns} +\end{frame} + +\begin{frame}[c] + \centering + \alert{How to find \only<1,2>{the best}\only<3>{\sout{the best} a good} elimination ordering?}\\ + + \bigskip + + \only<2>{ + \begin{align*} + \text{Best} &= G^+_\pi~\text{with Min(MAXCLIQUE}(G^+_\pi))\\ + &= \text{Computational Infeasible}\\ + &= \text{see}~\cite{heggernes2006} + \end{align*} + } + + \only<3>{ + \small + No best. But the smaller the triangulation the better.\\ + For minimal (not minimum): $\mathcal{O}(n^{2.376})$~\cite{heggernes2006} + } +\end{frame} + +\subsection{Greedy Triangulation} +\begin{frame}{Greedy Triangulation - Algorithm} + \begin{algorithm}[H] + \label{alg:greedy} + \KwIn{$G(V,E)$} + \KwOut{$\pi$} + $H=G$\\ + \For{$i=1$ \KwTo $n$}{ + Choose $v \in H$ by criterion \alert<2>{X}\\ + Set $\pi^{-1}(i) = v$\\ + Eliminate $v$ from $H$ (make $N_H(v)$ a clique and remove $v$) + } + \KwRet{H} + \end{algorithm} + \bigskip + + \uncover<2>{\centering \alert{How to choose X?}} +\end{frame} + +\begin{frame}{Greedy Triangulation - Criterion X} + \metroset{block=transparent} + \begin{block}{Minimum Degree/Greedy Degree} + X = $v$ with smallest degree in $H$\\ + \medskip + + {\small Performs well in practice} + \end{block} + + \begin{block}{Greedy Fill In} + X = $v$ which causes smallest number of fill edges in $G^+_\pi$\\ + \hspace{1.8mm} = $v$ with smallest number of pairs of non-adjacent neighbours\\ + \medskip + + {\small Slightly slower, slightly better bounds than MD/GD on average} + \end{block} + +\end{frame} + +\begin{frame}{Greedy Triangulation - Advanced Criteria} + \metroset{block=transparent} + \begin{block}{Lower Bound Based} + Eliminate $v$ from $H$, compute lower bound (LB) of treewidth\\ + Choose $v$ with Min($2*LB+\deg_H(v))$ + \end{block} + + \begin{block}{Enhanced Minimum Fill In} + Compute LB of $G$\\ + Choose simplical or almost simplical $v$ with $\deg(v)$ at most LB\\ + otherwise: Greedy Fill In + \end{block} + + \dots{} +\end{frame} + +\subsection{Local Search (Tabu Search)} + +\begin{frame}[shrink]{Tabu Search} + \begin{block}{General Approach} + \begin{enumerate}[(i)] + \item Keep list of $\alpha$ last solutions to avoid cycling + \item Find inital solution [= some elimination ordering] + \item Make small change to get \emph{Neighbourhood} + \item Select neighbouring solution $\not\in \alpha$ with smallest cost + \item Repeat (iii), (iv) some time $\rightarrow$ return best solution + \end{enumerate} + \end{block} + + \metroset{block=transparent} + \begin{block}{Neighbourhood Generation} + Swap two vertices in elminiation ordering + \end{block} + \begin{block}{Step Cost} + \begin{enumerate}[(i)] + \item Width of generated neighbour + \item But many neighbours with equal width, better: + $\rightarrow w_\pi * n^2 + \sum{v\in V}\vert N^+_\pi(v) \vert{}^2$ + \end{enumerate} + \end{block} + +\end{frame} + + +\subsection{Chordal Graph Recognition} + +\begin{frame}{Chordal Graph Recognition Heuristics} + If it's chordal already, find perfect elminiation ordering (i.e. recognize it): + \begin{itemize} + \item Maximum Cardinality Search + \item Lexicographical Breadth First Search + \end{itemize} + \dots{} tree decomposition depends on (perfect) elimination ordering found. Mostly determined by algorithms, except for first chosen $v_n$ (from right to left). + + $\rightarrow$ try for all $v$ + + $\rightarrow$ adds factor $\mathcal{O}(n)$ +\end{frame} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "../upperbounds" +%%% End: + +\message{ !name(../upperbounds.tex) !offset(-221) } + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/inputs/elimination.tex b/inputs/elimination.tex index 5af27f5..c2e99ad 100644 --- a/inputs/elimination.tex +++ b/inputs/elimination.tex @@ -107,8 +107,8 @@ \item $G^+_\pi$ is chordal \item $G$ is a subgraph of $G^+_\pi$ \item $\pi$ is a perfect elimination ordering of $G^+_\pi$ - \item $tw$ of \emph{subtree graph} (also a tree decomposition) of $G^+_\pi$ is $\text{MAXCLIQUE}(G^+_\pi)-1$ ~\cite{gavril1974} - \item There is a tree decomposition algorithm for $G$ with $tw = \text{MAXCLIQUE}(G^+_\pi)-1$, polynomial in n ~\cite{bodlaender2010} + \item $width$ of \emph{subtree graph} (also a tree decomposition) of $G^+_\pi$ is $\text{MAXCLIQUE}(G^+_\pi)-1$ ~\cite{gavril1974} + \item There is a tree decomposition algorithm for $G$ with $width = \text{MAXCLIQUE}(G^+_\pi)-1$, polynomial in n ~\cite{bodlaender2010} \end{itemize} } @@ -145,7 +145,7 @@ \KwOut{$\pi$} $H=G$\\ \For{$i=1$ \KwTo $n$}{ - Choose $v \in H$ by criterion \alert<2>{X}\\ + Choose $v \in H$ by criteria \alert<2>{X}\\ Set $\pi^{-1}(i) = v$\\ Eliminate $v$ from $H$ (make $N_H(v)$ a clique and remove $v$) } @@ -156,7 +156,7 @@ \uncover<2>{\centering \alert{How to choose X?}} \end{frame} -\begin{frame}{Greedy Triangulation - Criterion X} +\begin{frame}{Greedy Triangulation - Criteria X} \metroset{block=transparent} \begin{block}{Minimum Degree/Greedy Degree} X = $v$ with smallest degree in $H$\\ diff --git a/inputs/results.tex b/inputs/results.tex new file mode 100644 index 0000000..b738453 --- /dev/null +++ b/inputs/results.tex @@ -0,0 +1,20 @@ +\begin{frame}{Greedy Results} + + \begin{itemize} + \small + \item Average of 50 randomly generated graphs + \item Combinations of GreedyFillIn, GreedyDegree, Triangulation Minimisation + \item Best Results for combinations with Triangulation Minimisation + \item Worst Results for GreedyFillIn alone + \item GreedyDegree is fast and perfoming well + \end{itemize} + + \begin{figure} + \includegraphics[width=\linewidth]{res/results} + \caption{Results for Greedy Heuristics~\cite{bodlaender2010}} + \end{figure} +\end{frame} +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "../upperbounds" +%%% End: diff --git a/inputs/separators.tex b/inputs/separators.tex index 0913d97..429a4ff 100644 --- a/inputs/separators.tex +++ b/inputs/separators.tex @@ -4,6 +4,86 @@ \tableofcontents[sections={3}] \end{frame} +\subsection{Minimum Separating Vertex Set Heuristic} +\begin{frame}{Minimum Separating Vertex Set Heuristic} + +\metroset{block=transparent} +\begin{block}{} + + \begin{columns} + + \begin{column}{0.5\textwidth} + \begin{tikzpicture} + \node[shape=circle,draw=black] (A) at (0,0) {$X_{j_3}$}; + \node[shape=circle,draw=black] (B) at (3,0) {$X_{j_4}$}; + \node[shape=circle,draw=black] (C) at (1.5,1.5) {$X_i$}; + \only<2->{\node[shape=circle,fill=mLightBrown] (C) at (1.5,1.5) {$X_i$};} + \node[shape=circle,draw=black] (D) at (0,3) {$X_{j_1}$}; + \node[shape=circle,draw=black] (E) at (3,3) {$X_{j_2}$}; + + \path (A) edge (C); + \path (B) edge (C); + \path (D) edge (C); + \path (E) edge (C); + \draw[style=dashed] (E) -- (3,4); + \draw[style=dashed] (E) -- (3.8,3.8); + \draw[style=dashed] (E) -- (4,3); + \end{tikzpicture} + \end{column} + \only<3>{ + \begin{column}{0.5\textwidth} + \begin{tikzpicture} + \node[shape=circle,fill=mLightBrown] (C) at (0,0) {$S$}; + \node[shape=circle,fill=mLightBrown] (C1) at (-1,1) {$S \cup W_1$}; + \node[shape=circle,draw=black] (C11) at (-2,2) {$X_{j_1}$}; + + \node[shape=circle,fill=mLightBrown] (C2) at (1,1) {$S \cup W_2$}; + \node[shape=circle,draw=black] (C22) at (2,2) {$X_{j_2}$}; + + \node[shape=circle,fill=mLightBrown] (C3) at (-1,-1) {$S \cup W_3$}; + \node[shape=circle,draw=black] (C33) at (-2,-2) {$X_{j_3}$}; + + \node[shape=circle,fill=mLightBrown] (C4) at (1,-1) {$S \cup W_4$}; + \node[shape=circle,draw=black] (C44) at (2,-2) {$X_{j_4}$}; + + \draw (C) -- (C1) -- (C11); + \draw (C) -- (C2) -- (C22); + \draw (C) -- (C3) -- (C33); + \draw (C) -- (C4) -- (C44); + \draw[style=dashed] (C22) -- (2,3); + \draw[style=dashed] (C22) -- (2.8,2.8); + \draw[style=dashed] (C22) -- (3,2); + \end{tikzpicture} + \end{column} + } + \end{columns} +\end{block} + +\only<2>{ + \begin{block}{} + Choose $i\in I$ such that $\vert X_i \vert$ maximal and $G[X_i]$ does not include a clique. + \end{block} +} + +\only<3>{ + \begin{block}{} + Construct Graph $H_i$:\\ + $H_i(X_i,E_{H_i}), E_{H_i} = \{\{v,w\} \in X_i\times X_i \vert \{v,w\} \in E \vee \exists j \neq i: v,w \in X_j\}$\\ + Compute minimum separator $S$; $W_1,\dots{},W_r$ are components\\ + Construct new tree decomposition + \end{block} +} + +\end{frame} + +\subsection{Other Algorithms} +\begin{frame}{Others} + \begin{itemize} + \item MinimalTriangulation (same principle as Minimum Separating Vertex Set Heuristic) + \item Component Splitting + \end{itemize} +\end{frame} + %%% Local Variables: %%% mode: latex %%% TeX-master: "../upperbounds" diff --git a/res/results.png b/res/results.png new file mode 100644 index 0000000000000000000000000000000000000000..4f58655ac3ccd701491e1b753c3eb128b4ccaa13 GIT binary patch literal 205850 zcmcG$hd-9>{|77$B~b{;sw7Evl8h3G?97ntvW1L{N+o1xWhL305E)5Q3CRkfNLCV8 zS2oZ4?Ee0q=P!8f@3+@|XI$5Lp2z3-eBSH00yWeWcTlrZQ&3RsP&y~CMM1G?nu21( z)va6bm6xx+xZ_`2%+4#yQ>+vJdt8$lg|BRLI%nWQL9v~I__KlHQ6dYzNad=eszCL1 z3;S+58uQCbb@&qdRc(FOGmiH5mJY7?D+R@wYnG<2mKI#@S6#1gDJrRISO{2BQBZJE zD9NAJ_UQXL;Q7$VGJav!ao~pSP*v`{U6SixmaP+aT)9qty>ov*_4^Zvdsv)bZeVoX z*}iA@?nU9y?ULhPdHcutxo&T4{Tg?F%W1ANTb16P<=STO_S3|l!Los>3J0ClwYSxU zU8-D<(-<0sv z2X2b7oS4}W-aNWBpn>IH>X4p5>|v(hhsp=}qNypqIkD`!vs3PV@!7^xjb^(j>V^u< 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