init

slides_SPPCTW_elem.tex

+\newcommand{\lueckeColorless}[2]{\visible<#1|trans:0>{#2}}
+\newcommand{\lueckeOnlyColorless}[2]{\only<#1|trans:0>{#2}}
+
+
+
+
+\frame{
+\frametitle{SPPRCs -- relaxation of elementarity: $ng$-paths}\small
+\vspace*{-2ex}
+\begin{columns}[T]
+\begin{column}{0.47\textwidth}
+\textbf{ESPPRC:}
+\setlength{\leftmargini}{3.5ex}
+\begin{itemize}
+	\item[$\to$] Only elementary paths
+	\item[\green{$+$}] Strong LP-relaxation of the RMP
+	\item[\red{$-$}] Hard (sub)problem
+\end{itemize}
+\end{column}
+\hspace*{-6ex}
+\begin{column}{0.54\textwidth}
+\textbf{SPPRC:}
+\setlength{\leftmargini}{3.5ex}
+\begin{itemize}
+	\item[$\to$] Paths with arbitrary cycles
+	\item[\green{$+$}] Easier (sub)problem
+	\item[\red{$-$}] Weaker LP-relaxation of the RMP
+\end{itemize}
+\end{column}
+\end{columns}
+
+\bigskip\pause
+\textbf{SPPRC with $ng$-paths \citep{BaldacciEtAl2011}:}
+\begin{itemize}
+	\item \blue{Idea:} prohibit/allow certain cycles to handle trade-off between strength of the relaxation and difficulty of the subproblem
+	\item A \blue{neighborhood} $N_i\subset V\setminus\{s,t\}$ is associated with each node $i\in V$, e.g.~$i$ and its nearest neighbors
+	\item A \blue{cycle with a node $i$} is allowed only if the circle contains a node $j$ with $i\notin N_j$
+	\item \blue{Special cases of SPPRC with $ng$-paths}:\\
+				$\cTo$ $N_i = N$ for all $i$ $\Rightarrow$ \blue{ESPPRC}\\
+				$\cTo$ $N_i = \{i\}$ for all $i$ $\Rightarrow$ \blue{SPPRC}\\
+\end{itemize}
+%\vfill\vfill\vfill
+}
+
+\frame{
+\frametitle{SPPRCs -- relaxation of elementarity: $ng$-paths}\footnotesize
+\textbf{Example:}
+\tikzstyle myDepot=[minimum size = 0.35cm, draw=black, fill = red!95!black!55]
+\tikzstyle myCust=[circle=0.3pt, draw=black, fill = gray!50]
+\tikzstyle myLine=[color=cyan,line width=1pt, ->]
+\tikzstyle myLine2=[color=black,line width=1pt, ->]
+\hspace*{-3ex}
+\begin{tikzpicture}[]
+\node[myCust] (1) at (0,0) {1};	
+\node[myCust] (2) at (2,-2) {2};	
+\node[myCust] (3) at (4,0) {3};	
+\node[myCust] (4) at (2,2) {4};
+\node[myCust] (5) at (6,-2) {5};
+\node[myCust] (6) at (7,1) {6};
+\node[] (a) at (-2,-2) {};
+\node[] (b) at (-2,2) {};
+\node[] (c) at (-2,-1) {};
+\node[] (d) at (-2,1) {};
+\node[color=black,text width=1.5cm] (infeas) at (-2,0) {\phantom{infeasible cycle}};
+\lueckeColorless{2-}{
+\draw[color=mygreen, very thick, dashed](2,0) circle (2.5);
+\node[color=mygreen] (N1N4) at (2,2.7) {$N_1=N_4$};}
+\lueckeColorless{3-}{
+\draw[color=red, very thick, rotate around={45:(4.75,-0.75)},dashed](4.75,-0.75) ellipse (4 and 2.5);
+\node[color=red] (N5N6) at (7,-3) {$N_5=N_6$};}
+\lueckeColorless{4-}{
+\draw[color=blue, very thick,dashed](3,-1) ellipse (4 and 2.2);
+\node[color=blue] (N2N3) at (0,-3) {$N_2=N_3$};}
+\lueckeOnlyColorless{5}{
+\foreach \x/\y in {c/1, 1/2, 2/3, 3/4, 4/1, 1/d}
+\draw[myLine2] (\x) -- (\y);
+\node[color=black,text width=1.5cm] (infeas) at (-2,0) {infeasible cycle};}
+\lueckeOnlyColorless{6}{
+\foreach \x/\y in {a/1, 1/2, 2/6, 6/3, 3/1, 1/b}
+\draw[myLine2] (\x) -- (\y);
+\node[color=black,text width=1.5cm] (feas) at (-2,-2.5) {feasible cycle};}
+\end{tikzpicture}
+\footnotesize For clarity only, $N_1=N_4$, $N_2=N_3$ and $N_5=N_6$ have been chosen here.
+}
+