Skip to content
Snippets Groups Projects
Commit f45449b5 authored by Christian Tilk's avatar Christian Tilk
Browse files

init

parents
No related branches found
No related tags found
No related merge requests found
figure/m1.png

38.7 KiB

figure/m1n.png

20.8 KiB

figure/m2.png

39.4 KiB

figure/m2n.png

22.5 KiB

figure/m3.png

40.2 KiB

figure/m3n.png

23.4 KiB

figure/ng1.png

774 KiB

figure/ng2.png

301 KiB

figure/rf_clf1_roc.png

22.3 KiB

figure/rf_clf2_roc.png

22.3 KiB

figure/truck_loading.png

80.4 KiB

\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.
}
0% Loading or .
You are about to add 0 people to the discussion. Proceed with caution.
Please register or to comment