elimination completed

inputs/elimination.tex

@@ -7,12 +7,232 @@
 
 \subsection{Idea}
 
-\begin{frame}{}
+\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}
 
-\subsection{Test1}
+\begin{frame}{Greedy Triangulation - Advanced Criteria}
-\subsection{Test2}
+  \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

inputs/introduction.tex

@@ -1,8 +1,3 @@
-\begin{frame}{Table of contents}
-  \setbeamertemplate{section in toc}[sections numbered]
-  \tableofcontents[hideallsubsections]
-\end{frame}
-
 \section{Introduction}
 
 

inputs/motivation.tex

@@ -79,8 +79,8 @@
 
 \begin{frame}{Why Upper Bounds?}
 
-  \metroset{block=fill}
 
+  \metroset{block=fill}
   \begin{block}{We want (efficiently)}
     \begin{itemize}
     \item \emph{High} Lower Bound: Tree dec. not the right tool

inputs/separators.tex

@@ -0,0 +1,10 @@
+\begin{frame}{Table of contents}
+  \setbeamertemplate{section in toc}[sections numbered]
+  \setbeamertemplate{subsection in toc}[square]
+  \tableofcontents[sections={3}]
+\end{frame}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "../upperbounds"
+%%% End:

upperbounds.bib

@@ -1,5 +1,5 @@
 % This file was created with JabRef 2.10.
-% Encoding: ISO8859_1
+% Encoding: UTF8
 
 
 @Article{bodlaender2010,
@@ -18,12 +18,50 @@
   Keywords                 = {Treewidth}
 }
 
+@Article{gavril1974,
+  Title                    = {The intersection graphs of subtrees in trees are exactly the chordal graphs},
+  Author                   = {Fǎnicǎ Gavril},
+  Year                     = {1974},
+  Doi                      = {http://dx.doi.org/10.1016/0095-8956(74)90094-X},
+  ISSN                     = {0095-8956},
+  Number                   = {1},
+  Pages                    = {47 - 56},
+  Url                      = {http://www.sciencedirect.com/science/article/pii/009589567490094X},
+  Volume                   = {16},
+
+  __markedentry            = {[armin:]},
+  Abstract                 = {The intersection graph of a family of subtrees in an undirected tree is called a subtree graph. A graph is called chordal if every simple circuit with more than three vertices has an edge connecting two non-consecutive vertices. In this paper, we prove that, for a graph G, the following conditions are equivalent: (i) G is a chordal graph; (ii) G is a subtree graph; (iii) G is a proper subtree graph. Consider a chordal graph G. We give an efficient algorithm for constructing a representation of G by a family of subtrees in a tree.},
+  Journal                  = {Journal of Combinatorial Theory, Series B},
+  Owner                    = {armin},
+  Timestamp                = {2016.06.28}
+}
+
+@Article{heggernes2006,
+  Title                    = {Minimal triangulations of graphs: A survey },
+  Author                   = {Pinar Heggernes},
+  Year                     = {2006},
+  Doi                      = {http://dx.doi.org/10.1016/j.disc.2005.12.003},
+  ISSN                     = {0012-365X},
+  Note                     = {Minimal Separation and Minimal Triangulation },
+  Number                   = {3},
+  Pages                    = {297 - 317},
+  Url                      = {http://www.sciencedirect.com/science/article/pii/S0012365X05006060},
+  Volume                   = {306},
+
+  __markedentry            = {[armin:6]},
+  Abstract                 = {Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms. },
+  Journal                  = {Discrete Mathematics },
+  Keywords                 = {Chordal graphs},
+  Owner                    = {armin},
+  Timestamp                = {2016.06.28}
+}
+
 @Mastersthesis{roehrig1998,
   Title                    = {{{Tree Decomposition: A Feasibility Study}}},
   Author                   = {Hein R{\"o}hrig},
-  Institution              = {Max-Planck-Institut für Informatik},
+  Institution              = {Max-Planck-Institut für Informatik},
   Year                     = {1998},
-  Location                 = {Saarbrücken, Germany},
+  Location                 = {Saarbrücken, Germany},
   Url                      = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.132.7594&rep=rep1&type=pdf},
 
   Owner                    = {armin},

upperbounds.tex

@@ -1,4 +1,4 @@
-\documentclass[10pt]{beamer}
+\documentclass[10pt,dvipsnames]{beamer}
 \usepackage{appendixnumberbeamer}
 \usepackage{booktabs}
 \usepackage[scale=2]{ccicons}
@@ -7,9 +7,18 @@
 \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}}
@@ -35,19 +44,8 @@
 \section{Elimination Ordering Methods}
 \input{inputs/elimination}
 
-\begin{frame}{Table of contents}
-  \setbeamertemplate{section in toc}[sections numbered]
-  \setbeamertemplate{subsection in toc}[square]
-  \tableofcontents[sections={2}]
-\end{frame}
-
-\subsection{Idea}
-
-
-\subsection{Test1}
-\subsection{Test2}
-
 \section{Separator Methods}
+\input{inputs/separators}
 
 \section{Results}