finished

_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:

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 $H$\\

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:

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"

upperbounds.tex

@@ -9,6 +9,7 @@
 \usepackage{scalerel}
 \usepackage[normalem]{ulem}
 \usepackage[noend]{algorithm2e}
+\usepackage{adjustbox}
 
 % == General Beamer Settings ==
 \usetheme{metropolis}
@@ -20,10 +21,13 @@
 \setmathsfont{Oldstyle}
 
 
+
 % == Presentation Settings ==
 \def\thumbsup{\scalerel*{\includegraphics{res/up.png}}{O}}
 \def\thumbsdown{\scalerel*{\includegraphics{res/down.png}}{O}}
 
+\usetikzlibrary{graphs}
+
 \subject{Upper Bounds}
 \title[Upper Bounds]{Treewidth computations I. Upper bounds}
 \subtitle{Hans L. Bodlaender, Arie M.C.A Koster}
@@ -48,6 +52,7 @@
 \input{inputs/separators}
 
 \section{Results}
+\input{inputs/results}
 
 \begin{frame}[allowframebreaks]{References}
   \nocite{bodlaender2010}
@@ -55,6 +60,45 @@
   \bibliographystyle{abbrv}
 \end{frame}
 
+
+\begin{frame}[standout]
+  \begin{tikzpicture}
+    \node {Thanks!}
+    [grow=up]
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+    }
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+    }
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+    }
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}    
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+    }
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+    }
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+    }
+    child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)
+      child [draw=white, fill=white, shape=circle] {[fill] circle (0.2em)}
+    };
+  \end{tikzpicture}
+\end{frame}
+
 \end{document}
 
 %%% Local Variables: