\begin{frame}{General Motivation}
  \begin{columns}[c]
    \begin{column}{0.5\textwidth}
      \begin{enumerate}
      \item Choose infeasible problem
        \begin{itemize}
        \item Combinatorial Problems
        \item Computational Biology
        \item Constraint Satisfaction
        \item \dots{}
        \end{itemize}
      \item Find FPT$_{tw}$ algorithm
      \item Model problem as graph
      \item Compute \emph{tree composition} with small \emph{tree width}
      \uncover<2>{\alert{\item Tract the intractable}}
      \end{enumerate}
    \end{column}

    \begin{column}{0.8\textwidth}
      \begin{center}
      \begin{tikzpicture}[xscale=1.5,yscale=1.6]
        \tikzstyle{every node}=[]
        \tikzstyle{every path}=[line width=0pt]
        % 
        \begin{scope}
        \draw[fill,green,opacity=1,line width=0]
        (-1.5,0) -- (1.5,0)
        (-1,0) .. controls (-.8,1) and (.8,1) .. (1,0);
        ;
        % 
        \draw[fill,green!50,opacity=1,line width=0]
        (-1,0) .. controls (-.8,1) and (.8,1) .. (1,0)
        (1.1,0) .. controls (1.1,2.5) and (-1.1,2.5) .. (-1.1,0)
        ;
        % 
        \draw[fill,red!50,opacity=1,line width=0]
        (1.1,0) .. controls (1.1,2.5) and (-1.1,2.5) .. (-1.1,0)
        (-1.2,0) .. controls (-1.2,3.5) and (1.2,3.5) .. (1.2,0)
        ;
        % 
        \draw[fill,red!100,opacity=1,line width=0]
        (-1.2,0) .. controls (-1.2,3.5) and (1.2,3.5) .. (1.2,0)
        (1.5,0) .. controls (1.5,4.5) and (-3,4.5) .. (-1.5,0)
        ;
        \draw[fill,red!100,opacity=1,line width=0]
        (-1.2,0) .. controls (-1.2,3.5) and (1.2,3.5) .. (1.2,0)
        (1.5,0) .. controls (3,4.5) and (-1.5,4.5) .. (-1.5,0)
        ;
        % 
        \draw
        (-1.5,0) -- (1.5,0)
        (0,0.25) node[] {P}
        (-1,0) .. controls (-.8,1) and (.8,1) .. (1,0)
        % 
        (0,1.25) node[] {FPT}
        (-1.1,0) .. controls (-1.1,2.5) and (1.1,2.5) .. (1.1,0)
        % 
        (0,2.15) node[] {W[i]}
        (-1.2,0) .. controls (-1.2,3.5) and (1.2,3.5) .. (1.2,0)
        % 
        (-1,2.9) node[] {XP}
        % (0,2.8) node[circle,black,fill,inner sep=0pt] {}
        % (0,2.6) node[circle,black,fill,inner sep=0pt] {}
        % (0,2.4) node[circle,black,fill,inner sep=0pt] {}
        (-1.5,0) .. controls (-3,4.5) and (1.5,4.5) .. (1.5,0)
        % 
        (1,2.9) node[] {paraNP}
        % (0,2.8) node[circle,black,fill,inner sep=0pt] {}
        % (0,2.6) node[circle,black,fill,inner sep=0pt] {}
        % (0,2.4) node[circle,black,fill,inner sep=0pt] {}
        (-1.5,0) .. controls (-1.5,4.5) and (3,4.5) .. (1.5,0)
        ;
        \end{scope}
      \end{tikzpicture}
      \end{center}
    \end{column}
  \end{columns}  
\end{frame}

\begin{frame}{Why Upper Bounds?}

  \metroset{block=fill}

  \begin{block}{We want (efficiently)}
    \begin{itemize}
    \item \emph{High} Lower Bound: Tree dec. not the right tool
    \item \emph{Low} Upper Bound: Tree dec. works
    \item Other combinations? -- Not so useful
    \end{itemize}
  \end{block}

  \begin{block}{What this paper is about}
    \smallskip
    Exact algorithm: Huge constant factor \cite{roehrig1998}
    \begin{itemize}
    \item[$\rightarrow$] Find a non-optimal tree decomposition
    \item[$\rightarrow$] This is also an Upper Bound
    \end{itemize}
  \end{block}

\end{frame}

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