diff --git a/Main.tex b/Main.tex
index 80b7cfca832467a3ff5b0527d939ea2a90d157d9..129ed193c4031714e5b771657e4788e29baa4274 100644
--- a/Main.tex
+++ b/Main.tex
@@ -170,6 +170,16 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Titel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
+\tikzset{
+ mynode/.style={ellipse, draw=black, align=center, fill=yellow!40,very thick, text centered, minimum width={40pt},minimum height={30pt}},
+ mynode2/.style={rectangle, align=center,draw=black, fill=green!40,very thick, text centered, minimum width={75pt},minimum height={15pt}},
+ mynode3/.style={rectangle,align=center, draw=black, fill=green!40,very thick, text centered, minimum width={180pt},minimum height={12pt}},
+ mynodeB/.style={ellipse, draw=red, align=center, fill=yellow!40,very thick, text centered, minimum width={40pt},minimum height={30pt}},
+ edge/.style = {->,> = latex',thick},
+ mynode3B/.style={rectangle,align=center, draw=red, fill=green!40,very thick, text centered, minimum width={200pt},minimum height={15pt}},
+% myarrow/.style={->, >=latex', shorten >=1pt, thick},
+% mylabel/.style={text width=7em, text centered}
+}
%
\setbeamercovered{invisible}
%Standardkompiler txs:///pdflatex | txs:///bibtex | txs:///pdflatex | txs:///pdflatex
@@ -189,20 +199,215 @@
%\date{\AdvanceDate[1]\today}
\titlepage
}
-
-
- \frame{
+
+
+ \frame{
\frametitle{Supervised Learning}
-Learning: Pic from Proposal translated in englisch
+ \begin{columns}[T]
+ \begin{column}{0.68\textwidth}
+ \phantom{a}\\\medskip
+ \tikzstyle TestEdge=[-Triangle,line width=3pt,shorten >=3pt,green!40!black,dotted]
+\tikzstyle TrainEdge=[-Triangle,thick,shorten >=3pt,line width=3pt]
+\tikzset{
+ Legend/.style={draw=white, top color=white, bottom color=white, inner sep=0.5em, minimum height=1cm,text width=10mm, align = left}}
+\scalebox{0.7}{
+\begin{tikzpicture}[->, node distance=0.8cm, auto]
+\small
+\node[Legend] (Inst) {};%Trainingsdatenpunkte $i=1 \dots n$
+\node[mynode,below left=-1cm and 0.5cm of Inst] (Feat) {Input $X_1, \dots, X_n$};%Feature-Vektoren
+\node[mynode,below right=-1cm and 0.5cm of Inst] (Out) {Output $Y_1, \dots, Y_n$};
+\node[mynode,below=1.5cm of Inst,align=center] (ML) {ML-Model };
+\node[mynode,below=4cm of Inst,align=center] (Pred) {Trained ML-Model};
+
+\node[mynode,below left =2 cm and 0.5cm of Pred] (TestFeat) {Input $\hat{X}_1,, \dots, \hat{X}_m$};
+\node[mynode,below right=2 cm and 0.5cm of Pred] (OutTest) {Output $Y^*_1,\dots, Y^*_m$};
+
+\only<2>{
+\node[mynodeB,below left=-1cm and 0.5cm of Inst] (Feat) {\textbf{Input $X_1, \dots, X_n$}};%Feature-Vektoren
+\node[mynodeB,below right=-1cm and 0.5cm of Inst] (Out) {\textbf{Output $Y_1, \dots, Y_n$}};
+}
+\only<3>{
+\node[mynodeB,below=1.5cm of Inst,align=center] (ML) {\textbf{ML-Model} };
+}
+\only<4>{
+\node[mynodeB,below left =2 cm and 0.5cm of Pred] (TestFeat) {Input $\hat{X}_1,, \dots, \hat{X}_m$};
+}
+\only<5>{
+\node[mynodeB,below right=2 cm and 0.5cm of Pred] (OutTest) {Output $Y^*_1,\dots, Y^*_m$};
+}
+\path
+%(TestInst) edge[TestEdge] (TestFeat)
+(TestFeat) edge[TestEdge] (Pred)
+(Pred) edge[TestEdge] (OutTest)
+;
+
+\path
+%(Inst) edge[TrainEdge] node {} (Out)
+%(Inst) edge[TrainEdge] node {} (Feat)
+(Feat) edge[TrainEdge] node {}(ML)
+(Out) edge[TrainEdge] node {}(ML)
+(ML) edge[TrainEdge] node {}(Pred)
+;
+\node[Legend, below left = 1.5cm and 1.5cm of Inst] (E1) {Training};
+\node[Legend, below left= 2.5cm and 1.5cm of Inst] (E2) {Test};
+\node[Legend, left = 1cm of E1] (S1) {};
+\node[Legend, left = 1cm of E2] (S2) {};
+
+\path
+(S1) edge[TrainEdge] (E1)
+(S2) edge[TestEdge] (E2);
+\end{tikzpicture}
+}
+ \end{column}
+ \begin{column}{0.3\textwidth}
+ \begin{overprint}
+ \only<2>{
+ \includegraphics[width=\textwidth]{figure/catsdog2}}
+
+ \only<3>{
+ \includegraphics[width=\textwidth]{figure/GNN}
+ }
+ \only<4->{
+ \includegraphics[width=\textwidth]{figure/catsdog3}
+ \visible<5->{
+ \begin{center}
+ \Large \textbf{It's a dog!}
+ \end{center}
+ }
+ }
+ \end{overprint}
+ \end{column}
+ \end{columns}\pause\bigskip
+\large
+\begin{center}
+\visible<6->{
+\green{Works well for many classification and regression tasks!}
+}
+\end{center}
+ }
+
+\begin{frame}
+\frametitle{Operations Research}\footnotesize
+
+\begin{columns}[T]
+\begin{column}{.4\textwidth}
+\adjustbox{max height=0.9 \textheight}
+{
+ \begin{tikzpicture}
+
+\scriptsize
+\node[mynode] at (2,10.5) (1) {Problem};
+\node[mynode2] at (2,9) (2) {OR-expert};
+
+\node[mynode] at (2,7.5) (3a) {Mathematical \\ model};
+
+\node[mynode] at (2,6) (5a) {Algorithm};
+
+\visible<4->{
+\node[mynodeB] at (0.5,4) (5b) {\textbf{\normalsize{Data set}}};
+\node[mynodeB] at (3.5,4) (6) {\textbf{\normalsize{Solution}}};
+}
+\draw[edge] (1) -> (2);
+\draw[edge] (2) -> (3a);
+\draw[edge] (3a) -> (5a);
+\visible<4->{
+\draw[edge] (5a) -> (6);
+\draw[edge] (5b) -> (5a);
+}
+
+
+\onslide<1| trans:1>{
+\node[mynodeB] at (2,10.5) (a) {\normalsize{ \textbf{Problem}}};
+}
+\visible<2| trans:2>{
+\node[mynodeB] at (2,7.5) (b) { \textbf{\normalsize{Mathematical}} \\ \textbf{\normalsize{model}}};
+}
+\onslide<3| trans:3>{
+\node[mynodeB] at (2,6) (c) {\normalsize{\textbf{Algorithm}} };
+}
+%\onslide<4| trans:4>{
+% \node[mynodeB] at (3.4,5.5) (d) {\normalsize{\textbf{Data set}} \\};
+%}
+%\onslide<5| trans:5>{
+%\node[mynodeB] at (1.5,4.25) (e) {\normalsize{\textbf{Solution}}};
+%}
+
+
+\end{tikzpicture}
+}
+\end{column}
+\begin{column}{.6\textwidth}
+\begin{overprint}
+\onslide<1| trans:1>
+ \begin{bspBlock}{(Shortest path problem)}
+ Problem:\hfill\mbox{}\\[2ex]
+ \includegraphics[width=.99\textwidth]{figure/ZIMPL1}
+ \end{bspBlock}
+%
+\onslide<2| trans:2>
+ \begin{bspBlock}{(Shortest path problem)}
+ General mathematical model:\hfill\mbox{}\\[2ex]
+ \includegraphics[width=.99\textwidth]{figure/ZIMPL2}
+ \end{bspBlock}
+%
+\onslide<3| trans:3>
+ \begin{bspBlock}{(Dijkstra's Algorithm)}
+ \tiny
+ \begin{algorithm}[H]
+ \green{\tcp{Initialization}}
+ SET $d(s):= 0$, $pred(s):=undef$, $L:=\{s\}$\;
+ SET $d(j):=\infty$ for all $j\in V\setminus \{s\}$\;
+ \green{\tcp{Loop}}
+ \While{$L \ne \emptyset$}
+ {
+ SELECT $i\in L$ with $d(i)=\min_{k\in L} d(k)$\;
+ SET $L:=L\setminus\{i\}$~~\green{\tcp{$i$ is now ultimately marked}}
+ \For{$j: (i,j)\in\delta^+(i)$}
+ {
+ \If{$d(j)>d(i)+c_{ij}$}
+ {
+ \green{\tcp{Label}}
+ SET $d(j) := d(i)+c_{ij}$ and $pred(j):=i$\;
+ SET $L:=L \cup \{j\}$\;
+ }
+ }
+ }
+ \green{\tcp{Output: predecessors $pred(\cdot)$ and distances $d(\cdot)$}}
+\end{algorithm}
+ \end{bspBlock}
+ %
+%\onslide<4| trans:4>
+% \begin{bspBlock}{(Shortest path problem)}
+% \includegraphics[width=.99\textwidth]{figure/ZIMPL0}\\[3ex]
+% Data set (in .zpl file):\hfill\mbox{}\\[2ex]
+% \includegraphics[width=.99\textwidth]{figure/ZIMPL3}
+% \end{bspBlock}
+%
+%%
+%
+%%
+%
+%%
+\onslide<4| trans:4 >
+ \begin{bspBlock}{(Solution)}
+ \includegraphics[width=\textwidth]{figure/ZIMPL9b}
+ \end{bspBlock}
+
+\end{overprint}
+\end{column}
+\end{columns}
+\end{frame}
-works well for classification and regression tasks
+\frame{
+\frametitle{Machine Learning and Operations Research}
+\Large
+\textbf{Key Question:}\\
+\begin{center}
+\red{How can ML be used for solving optimisation problems?}\\[7ex]\pause
-Die Kernfrage
-ist, auf welche Weise ML in OR-Algorithmen eingesetzt werden
-kann, denn kombinatorische Optimierungsprobleme unterscheiden
-sich stark von den meisten Problemen, die aktuell durch ML gelst
-werden\citep{BengioEtAl2021}
+Combinatorial optimisation problems are quite different from most problems currently solved by ML \citep{BengioEtAl2021}
+\end{center}
}
\section{Vehicle Routing Problems}
@@ -383,13 +588,16 @@ Good/optimal solutions are often located in the border region.
\frame{
\frametitle{Combine ML with OR Algorithms!}
\small
-\blue{Apply learning inside OR Algorithms} \citep{BengioEtAl2021}:\bigskip
+ Sometimes expert knowledge is not satisfactory and algorithmic decisions are taken greedily or according to ``best practice''.\\\bigskip\pause
+ {\large
+\blue{$\Rightarrow:$ Apply learning inside OR Algorithms} \citep{BengioEtAl2021}:\medskip}\pause
+
\begin{itemize}
%\item End to end learning (only for little constraint problems like TSP)
-\item Learning to configure algorithms (expert knowledge is not satisfactory and the researcher wishes to find better ways of making decisions)
-\item Machine learning alongside optimization algorithms (e.g., call ML Model for each recurring decision in an optimization algorithm)
+\item Learning to configure algorithms (one-time decision)
+\item Machine learning alongside optimization algorithms (recurring decisions)
\end{itemize}
-\bigskip
+\bigskip\pause
Most ML Applications in OR focus on heuristics!
@@ -778,70 +986,32 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
%------------------------------------------------
-\newcommand{\RF}{RF}
-\newcommand{\modA}{DCH}
-\newcommand{\modB}{Hom}
-\newcommand{\modC}{Het}
+\newcommand{\RF}{\texttt{RF}}
+\newcommand{\modA}{\texttt{DCH}}
+\newcommand{\modB}{\texttt{Hom}}
+\newcommand{\modC}{\texttt{Het}}
\begin{frame}{Learning Models}
\small
Models used to perform the classification task:\\\medskip
\begin{itemize}
- \item[\RF] Random Forest with engineered variables
\item[\modA] Deep classification head of GNN encoded variables
- \item[\modB] Homegeneous GNN to encode NG-Set
- \item[\modC] Heterogeneous GNN to encode NG-Set
+ \item[\modB] Homogeneous GNN
+ \item[\modC] Heterogeneous GNN \pause
+ \item[\RF] Random Forest with engineered variables
\end{itemize}
+
% Additional attention layers have been omitted due to computing capacity and simplified models' inability to learn connections.
\end{frame}
%------------------------------------------------
-\begin{frame}{Random Forest Classifier Model}
-\small
-
-
- \textbf{Random Forest Classifier:}
- \begin{itemize}
- \item Uses log loss (cross-entropy) as the splitting criterion.
- \item Limits the depth of each tree to 100 levels to prevent overfitting.
- \item Each split considers at most 4 features, controlling tree diversity.
- \end{itemize}
-\end{frame}
%------------------------------------------------
-\begin{frame}{Encoding-Decoding GNN for Edge Prediction}
- \textbf{Graph Encoder:}
- \begin{itemize}
- \item \textbf{Graph Type:} Homogeneous (single node and edge type)
- \item \textbf{Convolution Layers:}
- \begin{itemize}
- \item \texttt{GCNConv1}: Projects input features to hidden space with ReLU activation.
- \item \texttt{GCNConv4}: Produces output node embeddings.
- \end{itemize}
- \item \textbf{Output:} Node embeddings used for edge prediction or passed to a classification model.
- \end{itemize}
-
- \textbf{Decoding Process:}
- \begin{itemize}
- \item Predicts edge scores using dot product of node embeddings.
- \item Computes full graph adjacency matrix for all possible node pairs.
- \end{itemize}
+\begin{frame}{Deep classification head of GNN encoded variables (\modA)}
+\textbf{Idea:} Encode nodes and classify with resulting node embeddings if ``$i$ is in neighborhood of $j$ or not''\\\pause\bigskip
- \textbf{Goal:} Predict edges or provide embeddings for downstream tasks.
-\end{frame}
-\begin{frame}{Classification Head for Node/Graph-Level Tasks}
- \textbf{Classification Model:}
- \begin{itemize}
- \item Fully connected feed-forward network for downstream tasks.
- \item \textbf{Structure:}
- \begin{itemize}
- \item 3 hidden layers with ReLU activations.
- \item \texttt{Input Size:} Embedding size from the encoder (e.g., 78).
- \item \texttt{Output:} Binary classification via sigmoid activation.
- \end{itemize}
- \end{itemize}
\textbf{Workflow:}
\begin{enumerate}
@@ -852,90 +1022,56 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
\item Combining graph-based features with external data.
\end{itemize}
\end{enumerate}
+ \end{frame}
+
+
+\begin{frame}{Homogeneous GNN (\modB)}
+\textbf{Idea:} Learn the arcs in the $ng$-graph directly\\\bigskip\pause
- \textbf{Goal:} Perform classification tasks using graph-based features.
-\end{frame}
-
-%------------------------------------------------
-
-\begin{frame}{Homogeneous GNN Model for Edge Prediction}
- \textbf{Architecture:}
+ \textbf{Architecture:}\\\smallskip
\begin{itemize}
\item \textbf{Graph Type:} Homogeneous (single node and edge type)
- \item \textbf{Convolution Layers:}
- \begin{itemize}
- \item 4 \texttt{GCNConv} layers for feature propagation and aggregation.
- \end{itemize}
- \item \textbf{Linear Layers:}
- \begin{itemize}
- \item Two optional linear transformations (\texttt{Linear1}, \texttt{Linear2}) for additional feature refinement.
+ \item 4 \blue{Convolution Layers} for feature propagation and aggregation (\texttt{GCNConv})
+ \item Two linear transformations for additional feature refinement
\end{itemize}
- \end{itemize}
-\end{frame}
+ \end{frame}
+%------------------------------------------------
-\begin{frame}{Homogeneous GNN Model for Edge Prediction II}
- \textbf{Encoding Process:}
- \begin{enumerate}
- \item Apply \texttt{GCNConv} layers sequentially:
- \begin{itemize}
- \item First 3 layers: Extract hierarchical features with ReLU activations.
- \item Final layer: Produces the output embeddings.
- \end{itemize}
- \item Optional: Dropout for regularization and edge dropout to sparsify training graphs.
- \end{enumerate}
-
- \textbf{Decoding Process:}
- \begin{itemize}
- \item Computes dot products between node embeddings to predict edge scores.
- \end{itemize}
- \textbf{Full Graph Prediction:}
- \begin{itemize}
- \item Computes pairwise similarities between all node embeddings for adjacency matrix reconstruction.
- \end{itemize}
- \textbf{Goal:} Predict edges in a homogeneous graph.
-\end{frame}
%------------------------------------------------
-\begin{frame}{Heterogeneous GNN Model for Edge Prediction}
- \textbf{Architecture:}
+\begin{frame}[noframenumbering]{Heterogeneous GNN (\modC)}
+
+\textbf{Idea:} Learn the arcs in the $ng$-gaph directly using different types of nodes and arcs\\\bigskip \pause
+ \textbf{Architecture:}\smallskip
\begin{itemize}
- \item \textbf{Node Types:} \texttt{stops}, \texttt{depot}
- \item \textbf{Edge Types:}
+ \item \textbf{Node Types:} \texttt{customer}, \texttt{depot}
+ \item \textbf{Arc Types:}
\begin{itemize}
- \item \texttt{route} (\texttt{stops} $\to$ \texttt{stops})
- \item \texttt{remember} (\texttt{stops} $\to$ \texttt{stops})
- \item \texttt{departs} (\texttt{depot} $\to$ \texttt{stops})
- \item \texttt{return} (\texttt{stops} $\to$ \texttt{depot})
+ \item \texttt{route} (\textit{customer} $\to$ \textit{customer})
+ \item \texttt{remember} (\textit{customer} $\to$ \textit{customer})
+ \item \texttt{departs} (\textit{depot} $\to$ \textit{customer})
+ \item \texttt{return} (\textit{customer} $\to$ \textit{depot})
\end{itemize}
- \item \textbf{Convolution Layers:} Uses \texttt{GCNConv} and \texttt{SAGEConv} for message passing.
+ \item \textbf{Convolution Layers:} (\texttt{GCNConv} and \texttt{SAGEConv})
\end{itemize}
\end{frame}
-\begin{frame}{Heterogeneous GNN Model for Edge Prediction II}
- \textbf{Encoding Process:}
- \begin{enumerate}
- \item Linear projections for \texttt{stops} and \texttt{depot} inputs.
- \item \textbf{Two-stage Convolution:}
- \begin{itemize}
- \item \textbf{Stage 1:} Combines \texttt{route}, \texttt{remember}, and \texttt{depot} connections using GCN/SAGE layers.
- \item \textbf{Stage 2:} Refines embeddings using updated node features.
- \end{itemize}
- \item Produces final embeddings for \texttt{stops} and \texttt{depot}.
- \end{enumerate}
- \textbf{Decoding Process:}
+
+
+\begin{frame}{Random Forest (\RF)}
+\small
+
\begin{itemize}
- \item Predicts sparse, directed \texttt{remember} edges using dot product of \texttt{stops} embeddings.
+ \item Uses log loss (cross-entropy) as the splitting criterion.
+ \item Limits the depth of each tree to 100 levels to prevent overfitting.
+ \item Each split considers at most 4 features, controlling tree diversity.
\end{itemize}
-
- \textbf{Goal:} Predict sparse directed edges between \texttt{stops}.
\end{frame}
-
-
%\begin{frame}{Random forest summary}
%\red{??? How to interpret?}
% \begin{columns}[c]
@@ -956,7 +1092,7 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
% \end{columns}
%\end{frame}
-\begin{frame}{Preliminary Results -- Summary}
+\begin{frame}{Preliminary Results -- \modA}
% \begin{table}
% \begin{tabular}{l l l}
% \toprule
@@ -969,44 +1105,37 @@ How can we find such a neighborhood?\\\smallskip\pause \blue{Dynamic Neighborhoo
% \end{tabular}
% \caption{Table caption}
% \end{table}
-
- \begin{columns}[c]
- \column{.5\textwidth}
- \begin{figure}[h]
- \centering
- \includegraphics[width=0.9\textwidth]{figure/rf_clf1_roc.png}
- \caption{\RF, Acc=0.923}
- \end{figure}
-
- \column{.5\textwidth}
- \begin{figure}[h]
-
- \centering
- \includegraphics[width=\textwidth]{figure/m1n.png}
- \caption{\modA, Acc=0.895}
- \end{figure}
-
- \end{columns}
+\begin{figure}[h]
+ \includegraphics[width=0.9\textwidth]{figure/m1n.png}
+\end{figure}\Large
+Accuracy 0.895
\end{frame}
-\begin{frame}{Preliminary Results -- Summary}
+\begin{frame}{Preliminary Results -- \modB~and \modC}
+\Large
\begin{columns}[c]
\column{.5\textwidth}
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figure/m2n.png}
- \caption{\modB, Acc=0.876}
+ \caption{\modB, Accuracy 0.876}
\end{figure}
\column{.5\textwidth}
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figure/m3n.png}
- \caption{\modC, Acc=0.785}
+ \caption{\modC, Accuracy 0.785}
\end{figure}
\end{columns}
\end{frame}
-
+\begin{frame}{Preliminary Results -- \RF}
+ \begin{figure}[h]
+ \centering
+ \includegraphics[height=0.8\textheight]{figure/rf_clf1_roc.png}
+ \caption{\RF, Accuracy 0.923}
+ \end{figure}
+ \end{frame}
\begin{frame}{DNE-neighborhood vs learned neighborhood}
@@ -1035,14 +1164,16 @@ Picture neighborhood vs learned neighborhood - are the learned rather sparse or
\begin{frame}{Conclusion}
\begin{itemize}\setlength\itemsep{1em}
- \item Result using the encoding head were rather disappointing
- \item (node-) encoding a complete graph seems to have disadvantages and some heuristics must be performed (e.g.,kNN).
+ \item Performance of GNNs rather disappointing (compared to \RF)
+ %\item (node-) encoding a complete graph seems to have disadvantages and some heuristics must be performed (e.g.,kNN).
\item Do we need more/other engineered variables?
- \item Heterogeneity in the nature of the graph hasn't shown increase in performance.
+ \item What are good attention layers?
+ \item Using heterogeneity in the nature of the graph has shown decrease in performance.
\end{itemize}\pause
\end{frame}
\begin{frame}{Outlook}
\begin{itemize}\setlength\itemsep{1em}
+ \item Evaluate importance of input features
\item Evaluate the learning success by using the learned neighborhood in a fully-fledged branch-price-and-cut algorithm.
\item Extend the research on other VRP-variants
\item Use Learning in other parts of the branch-price-and-cut algorithm
@@ -1057,7 +1188,9 @@ Picture neighborhood vs learned neighborhood - are the learned rather sparse or
Questions or remarks?!
\end{center}
}
-
+ \scriptsize
+ \bibliographystyle{apalike}
+ \bibliography{References}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1101,10 +1234,94 @@ Picture neighborhood vs learned neighborhood - are the learned rather sparse or
\end{tikzpicture}
\footnotesize For clarity only, $N_1=N_4$, $N_2=N_3$ and $N_5=N_6$ have been chosen here.
\end{frame}
- \scriptsize
- \bibliographystyle{apalike}
- \bibliography{References}
-
-
+
+ \begin{frame}[noframenumbering]{\modA}
+ \textbf{Graph Encoder:}
+ \begin{itemize}
+ \item \textbf{Graph Type:} Homogeneous (single node and edge type)
+ \item \textbf{Convolution Layers:}
+ \begin{itemize}
+ \item \texttt{GCNConv1}: Projects input features to hidden space with ReLU activation.
+ \item \texttt{GCNConv4}: Produces output node embeddings.
+ \end{itemize}
+ \item \textbf{Output:} Node embeddings
+ \end{itemize}
+
+% \textbf{Decoding Process:}
+% \begin{itemize}
+% \item Predicts edge scores using dot product of node embeddings.
+% \item Computes full graph adjacency matrix for all possible node pairs.
+% \end{itemize}
+
+ \textbf{Goal:} Predict edges or provide embeddings for downstream tasks.
+\end{frame}
+
+\begin{frame}[noframenumbering]{\modA}
+ \textbf{Classification Model:}
+ \begin{itemize}
+ \item Fully connected feed-forward network for downstream tasks.
+ \item \textbf{Structure:}
+ \begin{itemize}
+ \item 3 hidden layers with ReLU activations.
+ \item \texttt{Input Size:} Embedding size from the encoder (e.g., 78).
+ \item \texttt{Output:} Binary classification via sigmoid activation.
+ \end{itemize}
+ \end{itemize}
+
+ \textbf{Workflow:}
+ \begin{enumerate}
+ \item Encode node embeddings using GNN layers.
+ \item Pass embeddings to classification head for:
+ \begin{itemize}
+ \item Node/graph classification tasks.
+ \item Combining graph-based features with external data.
+ \end{itemize}
+ \end{enumerate}
+
+ \textbf{Goal:} Perform classification tasks using graph-based features.
+\end{frame}
+ \begin{frame}[noframenumbering]{Homogeneous GNN Model}
+ \textbf{Encoding Process:}
+ \begin{enumerate}
+ \item Apply \texttt{GCNConv} layers sequentially:
+ \begin{itemize}
+ \item First 3 layers: Extract hierarchical features with ReLU activations.
+ \item Final layer: Produces the output embeddings.
+ \end{itemize}
+ \item Optional: Dropout for regularization and edge dropout to sparsify training graphs.
+ \end{enumerate}
+
+ \textbf{Decoding Process:}
+ \begin{itemize}
+ \item Computes dot products between node embeddings to predict edge scores.
+ \end{itemize}
+
+ \textbf{Full Graph Prediction:}
+ \begin{itemize}
+ \item Computes pairwise similarities between all node embeddings for adjacency matrix reconstruction.
+ \end{itemize}
+
+ \textbf{Goal:} Predict edges in a homogeneous graph.
+\end{frame}
+
+\begin{frame}[noframenumbering]{Heterogeneous GNN Model}
+ \textbf{Encoding Process:}
+ \begin{enumerate}
+ \item Linear projections for \texttt{customer} and \texttt{depot} inputs.
+ \item \textbf{Two-stage Convolution:}
+ \begin{itemize}
+ \item \textbf{Stage 1:} Combines \texttt{customer}, \texttt{ng}, and \texttt{depot} connections using GCN/SAGE layers.
+ \item \textbf{Stage 2:} Refines embeddings using updated node features.
+ \end{itemize}
+ \item Produces final embeddings for \texttt{customer} and \texttt{depot}.
+ \end{enumerate}
+
+ \textbf{Decoding Process:}
+ \begin{itemize}
+ \item Predicts sparse, directed \texttt{ng} edges using dot product of \texttt{customer} embeddings.
+ \end{itemize}
+
+ \textbf{Goal:} Predict sparse directed edges between \texttt{customer}.
+\end{frame}
\end{document}
diff --git a/figure/GNN.jpg b/figure/GNN.jpg
new file mode 100644
index 0000000000000000000000000000000000000000..a8652b50c814f06c63482541b712720fa6274c44
Binary files /dev/null and b/figure/GNN.jpg differ
diff --git a/figure/ML_anw1.tex b/figure/ML_anw1.tex
new file mode 100644
index 0000000000000000000000000000000000000000..641bbdb2539749eee9191460923e467ade25f39d
--- /dev/null
+++ b/figure/ML_anw1.tex
@@ -0,0 +1,22 @@
+
+\begin{center}
+
+
+\begin{tikzpicture}[->, node distance=0.8cm, auto]
+\small
+\node[mynode] (Inst) {Testdatenpunkt};
+\node[mynode,below=0.5 cm of Inst] (Feat) {Feature-Vector};
+\node[mynode,below=0.5 cm of Feat] (ML) {Trainiertes ML-Verfahren};
+\node[mynode,below=0.5 cm of ML] (Out) {Vorhergesagter Output};
+\path
+(Inst) edge[-Triangle,thick,shorten >=3pt] (Feat)
+(Feat) edge[-Triangle,thick,shorten >=3pt] (ML)
+(ML) edge[-Triangle,thick,shorten >=3pt] (Out)
+;
+\end{tikzpicture}
+
+
+
+
+
+\end{center}
\ No newline at end of file
diff --git a/figure/ML_anw2.tex b/figure/ML_anw2.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7cf86b33952181e658cfd88b49b182a197031940
--- /dev/null
+++ b/figure/ML_anw2.tex
@@ -0,0 +1,27 @@
+
+\begin{center}
+
+
+
+
+\begin{tikzpicture}[->, node distance=0.8cm, auto]
+\small
+\node[large, below=1cm of Inst,text depth = 7 cm] (OR) {};
+\node[mynode, above = 1cm of OR] (Inst) {Testinstanz};
+\node[mynode,below right=-4.5cm and -9cm of OR] (Feat) {Feature-Vector};
+\node[mynode,below right=-4.5cm and -4cm of OR] (Out) {Output};
+\node[mynode,below =-7cm of OR] (Ver) {OR-Verfahren};
+\node[mynode,below =-2cm of OR] (ML) {Trainiertes ML-Verfahren};
+\node[mynode2,right =2cm of OR] (Los) {Lösung};
+
+\path
+(Inst) edge[thick] (OR)
+(Ver) edge[thick] (Feat)
+(Feat) edge[thick] (ML)
+(ML) edge[thick] (Out)
+(Out) edge[thick] (Ver)
+(OR) edge[thick] (Los)
+;
+\end{tikzpicture}
+
+\end{center}
\ No newline at end of file
diff --git a/figure/ML_train.tex b/figure/ML_train.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4dadbe7388555c5f699d154401877919bf7d4db9
--- /dev/null
+++ b/figure/ML_train.tex
@@ -0,0 +1,17 @@
+
+\begin{center}
+\begin{tikzpicture}[->, node distance=0.8cm, auto]
+\small
+
+\node[mynode] (Inst) {Trainingsdatenpunkt};
+\node[mynode,below left=1cm and -0.5cm of Inst] (Feat) {Feature-Vector};
+\node[mynode,below right=1cm and -0.5cm of Inst] (Out) {Berechneter Output};
+\node[mynode,below=3cm of Inst,align=center] (ML) {ML-Verfahren };
+\path
+(Inst) edge[-Triangle,thick,shorten >=3pt] node {} (Out)
+(Inst) edge[-Triangle,thick,shorten >=3pt] node {} (Feat)
+(Feat) edge[-Triangle,thick,shorten >=3pt] node {}(ML)
+(Out) edge[-Triangle,thick,shorten >=3pt] node {}(ML)
+;
+\end{tikzpicture}
+\end{center}
\ No newline at end of file
diff --git a/figure/ML_train_new.tex b/figure/ML_train_new.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1c1366f921c9863ba624ba53709839b8b2737fab
--- /dev/null
+++ b/figure/ML_train_new.tex
@@ -0,0 +1,44 @@
+\tikzstyle TestEdge=[-Triangle,line width=3pt,shorten >=3pt,green!40!black,dotted]
+\tikzstyle TrainEdge=[-Triangle,thick,shorten >=3pt,line width=3pt]
+\tikzset{
+ Legend/.style={draw=white, top color=white, bottom color=white, inner sep=0.5em, minimum height=1cm,text width=10mm, align = left}}
+
+\begin{tikzpicture}[->, node distance=0.8cm, auto]
+\small
+
+\node[Legend] (Inst) {};%Trainingsdatenpunkte $i=1 \dots n$
+\node[mynode,below left=-1cm and -0.5cm of Inst] (Feat) {Input $X_1, \dots, X_n$};%Feature-Vektoren
+\node[mynode,below right=-1cm and -0.5cm of Inst] (Out) {Output $Y_1, \dots, Y_n$};
+\node[mynode,below=1.5cm of Inst,align=center] (ML) {ML-Model };
+\node[mynode,below=4cm of Inst,align=center] (Pred) {Prediction};
+
+
+\node[Legend, above left = 1cm and 2.5cm of Pred] (TestInst) {}; %Testdatenpunkt $j$
+\node[mynode,below=1 cm of TestInst] (TestFeat) {Input $\hat{X}_1,, \dots, \hat{X}_m$};
+
+\node[mynode,right=1.5 cm of Pred] (OutTest) {Output $Y^*_1,\dots, Y^*_m$};
+\path
+%(TestInst) edge[TestEdge] (TestFeat)
+(TestFeat) edge[TestEdge] (Pred)
+(Pred) edge[TestEdge] (OutTest)
+;
+
+\path
+%(Inst) edge[TrainEdge] node {} (Out)
+%(Inst) edge[TrainEdge] node {} (Feat)
+(Feat) edge[TrainEdge] node {}(ML)
+(Out) edge[TrainEdge] node {}(ML)
+(ML) edge[TrainEdge] node {}(Pred)
+;
+
+
+
+\node[Legend, below right = 1cm and 5cm of Inst] (E1) {Training};
+\node[Legend, below = -0.25cm of E1] (E2) {Test};
+\node[Legend, left = 1cm of E1] (S1) {};
+\node[Legend, left = 1cm of E2] (S2) {};
+
+\path
+(S1) edge[TrainEdge] (E1)
+(S2) edge[TestEdge] (E2);
+\end{tikzpicture}
diff --git a/figure/OR_algo.tex b/figure/OR_algo.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1c1366f921c9863ba624ba53709839b8b2737fab
--- /dev/null
+++ b/figure/OR_algo.tex
@@ -0,0 +1,44 @@
+\tikzstyle TestEdge=[-Triangle,line width=3pt,shorten >=3pt,green!40!black,dotted]
+\tikzstyle TrainEdge=[-Triangle,thick,shorten >=3pt,line width=3pt]
+\tikzset{
+ Legend/.style={draw=white, top color=white, bottom color=white, inner sep=0.5em, minimum height=1cm,text width=10mm, align = left}}
+
+\begin{tikzpicture}[->, node distance=0.8cm, auto]
+\small
+
+\node[Legend] (Inst) {};%Trainingsdatenpunkte $i=1 \dots n$
+\node[mynode,below left=-1cm and -0.5cm of Inst] (Feat) {Input $X_1, \dots, X_n$};%Feature-Vektoren
+\node[mynode,below right=-1cm and -0.5cm of Inst] (Out) {Output $Y_1, \dots, Y_n$};
+\node[mynode,below=1.5cm of Inst,align=center] (ML) {ML-Model };
+\node[mynode,below=4cm of Inst,align=center] (Pred) {Prediction};
+
+
+\node[Legend, above left = 1cm and 2.5cm of Pred] (TestInst) {}; %Testdatenpunkt $j$
+\node[mynode,below=1 cm of TestInst] (TestFeat) {Input $\hat{X}_1,, \dots, \hat{X}_m$};
+
+\node[mynode,right=1.5 cm of Pred] (OutTest) {Output $Y^*_1,\dots, Y^*_m$};
+\path
+%(TestInst) edge[TestEdge] (TestFeat)
+(TestFeat) edge[TestEdge] (Pred)
+(Pred) edge[TestEdge] (OutTest)
+;
+
+\path
+%(Inst) edge[TrainEdge] node {} (Out)
+%(Inst) edge[TrainEdge] node {} (Feat)
+(Feat) edge[TrainEdge] node {}(ML)
+(Out) edge[TrainEdge] node {}(ML)
+(ML) edge[TrainEdge] node {}(Pred)
+;
+
+
+
+\node[Legend, below right = 1cm and 5cm of Inst] (E1) {Training};
+\node[Legend, below = -0.25cm of E1] (E2) {Test};
+\node[Legend, left = 1cm of E1] (S1) {};
+\node[Legend, left = 1cm of E2] (S2) {};
+
+\path
+(S1) edge[TrainEdge] (E1)
+(S2) edge[TestEdge] (E2);
+\end{tikzpicture}
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