init

JP pres/feb_presentation.tex

+%----------------------------------------------------------------------------------------
+%	PACKAGES AND THEMES
+%----------------------------------------------------------------------------------------
+\documentclass[aspectratio=169,xcolor=dvipsnames]{beamer}
+\usetheme{Madrid}
+
+\usepackage{hyperref}
+\usepackage{graphicx} % Allows including images
+\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[format=plain,justification=center]{caption}
+\usepackage{subcaption}
+
+%----------------------------------------------------------------------------------------
+%	TITLE PAGE
+%----------------------------------------------------------------------------------------
+
+% The title
+\title[Learning NG-Sets]{Learning the NG sets in Branch-and-Price strategies for routing problems}
+\subtitle{}
+
+\author[JP Cantillana] {Juan Pablo Cantillana}
+\institute[Business Analytics] % Your institution may be shorthand to save space
+{   
+    \begin{center}
+    \includegraphics[width=6cm]{Uni_Logo_blau.jpg}
+    \end{center}
+    % Your institution for the title page
+    Department of Business Analytics \\
+    University of Vienna 
+    \vskip 3pt
+}
+\date{\today} % Date, can be changed to a custom date
+
+\AtBeginSection[]
+{
+  \begin{frame}
+    \frametitle{Table of Contents}
+    \tableofcontents[currentsection]
+  \end{frame}
+}
+
+%----------------------------------------------------------------------------------------
+%	PRESENTATION SLIDES
+%----------------------------------------------------------------------------------------
+
+\begin{document}
+
+\begin{frame}
+    % Print the title page as the first slide
+    \titlepage
+    
+\end{frame}
+
+\begin{frame}{Overview}
+    % Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
+    \tableofcontents
+\end{frame}
+
+%------------------------------------------------
+\section{Research question}
+%------------------------------------------------
+
+% \begin{frame}{Research team}
+% \end{frame}
+
+%------------------------------------------------
+\subsection{Column generation}
+\begin{frame}{Column generation - Motivation}
+    \begin{itemize}
+        \item Sometimes when obtaining a solution from a Mixed Integer Problem is easy.
+        \item There are times when it's not the case, so reformulations must be thought.
+        \item Representation theorem allows enumerating extreme points and rays.
+        \item In many problems these points and rays can be obtained easily.
+    \end{itemize}
+    \begin{figure}
+        \centering
+        \includegraphics[width=0.4\linewidth]{corner2.png}
+        \caption{In the edge-based formulation, a route can be imagined as the corner of the shortest path hypercube as in the route $\{1,2,3,4,1\}$ on the left. Many routes can be vertices of the SP, and when multiple vehicles are allowed these can be aggregated and the solutions found as linear combinations of these corners/routes.}
+        \label{fig:cornerxij}
+    \end{figure}
+\end{frame}
+
+\begin{frame}{Column Generation}
+    \begin{itemize}
+        \item Column Generation (CG) exploits this ability.
+        \item We divide the problem in 2, where the constraints that define the corners (e.g. in routing, the arc-based routes) are going to be considered the Subproblem (SP), and the ones that evaluate how these are combined the Master or Main problem (MP).
+        \item The MP is solved using Simplex, can we reduce the number of (aggregated) variables?
+        \item We need to generate the reduced costs for solving the SP and produce variables in a coordinated way.
+    \end{itemize}
+    \begin{figure}[htb]
+        \centering
+            \centering
+            \includegraphics[width=0.25\linewidth]{cg1.png}
+            \includegraphics[width=0.25\linewidth]{cg2.png}
+            \caption{The method is called Column Generation because the MP is solved (using Simplex) in such a way that we will slowly introduce columns to the $A$ matrix to form the base, and out of the reduced costs (in the form of dual variables) we'll get the costs for solving the SP.}
+    \end{figure}
+\end{frame}
+
+%------------------------------------------------
+
+
+\begin{frame}{Column generation in VRP}
+    \begin{itemize}
+        \item VRP are a family of problems where the difficulty grows fast in the number of stops, so we use tools such as Column Generation to solve them to optimality.
+        \item The route-specific constraints that form the Shortest Path can also be solved by a Label Setting algorithm.
+    \end{itemize}
+    \begin{alertblock}{Relevant}
+        Different constraint relaxations can be used in order to produce solutions in a cheaper way. Many concentrate on the Subtour Elimination Constraints, but this sacrifices the elementarity (absence of cycles) in the tours.
+    \end{alertblock}
+    \begin{examples}
+        Methods that explore that relaxation are e.g. Subset row cuts, k-cycle elimination, partial elementarity and NG-Path relaxation.
+    \end{examples}
+\end{frame}
+
+%------------------------------------------------
+
+\begin{frame}{Column generation scheme}
+    \begin{figure}
+        \centering
+        \includegraphics[width=0.6\linewidth]{cg.png}
+        \caption{From Prof. Tilk's presentations. This scheme shows how Column Generation is used for routing problems.}
+        \label{fig:cornerxij}
+    \end{figure}
+\end{frame}
+
+%------------------------------------------------
+\subsection{NG-Sets}
+\begin{frame}{No-memory Label setting}
+    \begin{figure}
+        \centering
+        \includegraphics[width=0.5\linewidth]{memoryless.png}
+        \caption{We can produce easily routes, but if these contain cycles, we'll call these non-elementary (and infeasible). We can include a memory resource in the labeling for SP with resource constraints, but that's not so efficient. For this specific route, we should only remember node 1 but not node 4.}
+        \label{fig:memory}
+    \end{figure}
+\end{frame}
+
+\begin{frame}{NG-Sets}
+    \begin{itemize}
+        \item Ng-Path relaxation is a tool used for speeding-up elementarity in solutions obtained via SPPRC algorithm. These rely on the usage of Ng-Sets associated to each stop to compute a resource called the memory.
+        \item Usually it's defined over a vicinity of a given radius, or a set of the k-nearest neighbors.
+        \item The memory as a route resource is updated each time it's extended, but it's allowed to forget.
+        \item Also dominance checks on routes can be done by keeping the label with the smallest memory when all other resources are ceteris paribus.
+    \end{itemize}
+    
+\end{frame}
+
+%------------------------------------------------
+
+\begin{frame}{Learning the NG-Sets pic1}
+    \frametitle{Learning the NG-Sets}
+    \begin{figure}[h]
+            
+        \centering
+        \includegraphics[width=0.85\linewidth]{newplot3.png}
+        \caption{Example of Ng-Set as a directed graph for a simple instance.}
+    \end{figure}
+    
+\end{frame}
+\begin{frame}{Learning the NG-Sets pic2}
+    \frametitle{Learning the NG-Sets}
+    \begin{figure}[h]
+            
+        \centering
+        \includegraphics[width=0.85\linewidth]{newplot2.png}
+        \caption{Example of Ng-Set as a directed graph. Expansion around node 37.}
+    \end{figure}
+    
+\end{frame}
+
+%------------------------------------------------
+
+\begin{frame}{Learning the NG-Sets}
+    \begin{itemize}
+        \item The methods to obtain these Ng-Sets are mostly parameter-reliant, and the question on which shall be used to obtain an elementary root solution in a branching scheme is still open.
+        \item The Augmented Ng-Set or \textit{AugNg-Set} is a tool to determine the natural size of these Ng-Sets, but can it be learned?
+        \item Obtaining the data until the solution obtained in the root node is elementary is not excessively costly
+    \end{itemize}
+    \begin{alertblock}{Research hypothesis}
+        The Ng-relaxation induced lower bound can be learned out of the AugNg-Sets obtained in the solution of the root node.
+    \end{alertblock}
+\end{frame}
+
+%------------------------------------------------
+
+\section{Methods}
+
+%------------------------------------------------
+\subsection{Data acquisition}
+\begin{frame}{Obtaining instances}
+    \begin{itemize}
+        \item Instances created should generate a range of sizes for NG-Sets, and difficulty levels to be solved
+        \item For comparison reasons, space is limited in the 100x100 square
+        \item Variable parameters considered include:
+        \begin{itemize}
+            \item Degree of centrality of depot
+            \item Width of time windows
+            \item Number of clusters of stops
+            \item Distribution of stops
+            \item Capacity of vehicles
+        \end{itemize}
+        \item These form scenario families from which we can sample. In total over 20,000 instances are generated.
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Obtaining NG-Sets}
+    \begin{itemize}
+        \item We limit the study to the Aug-Ng-Set generated in the root relaxation.
+        \item Each instance is solved in the root node.
+        \item The Augmentation is performed until the solution obtained is elementary.
+        \item The obtained set is then stored and can be used for classification tasks.
+    \end{itemize}
+    \begin{columns}[c]
+        \column{.33\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=4cm]{aug1.png}
+            \caption{A non elementary column is generated.}
+        \end{figure}
+        \column{.33\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=4cm]{aug2.png}
+            \caption{We detect it and add the conflicting node to the NG-sets of the cycle so it's not forgotten}
+        \end{figure}
+        \column{.33\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=4cm]{aug3.png}
+            \caption{A new column is generated. As it's elementary, this provides no new information for augmentation.}
+        \end{figure}
+    \end{columns}
+\end{frame}
+
+\begin{frame}{Obtaining Attributes}
+    \begin{itemize}
+        \item General attributes, such as instance-based number of vehicles and capacity, or stop-related such as demand, location, time windows and service times are included.
+        \item Additionally, engineered variables are considered, including:
+        \begin{itemize}
+            \item Distance to the element furthest away from the depot
+            \item Average length to depot
+            \item Average route length as $\frac{totalDemand}{Capacity}$
+            \item Euclidean distance from each stop to the depot
+            \item Amount of remaining time to visit a given stop
+            \item Amount of remaining capacity to visit a given stop.
+        \end{itemize}
+        \item On top of that, also Graph Encoding variables of the kNN induced graph can be considered.
+    \end{itemize}
+\end{frame}
+
+%------------------------------------------------
+\subsection{Models}
+\begin{frame}{Models used}
+    Models used to perform the learning task are:
+    \begin{itemize}
+        \item Random Forest with engineered variables
+        \item Deep classification head of GNN encoded variables
+        \item Homegeneous GNN to encode NG-Set
+        \item Heterogeneous GNN to encode NG-Set
+    \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}
+    \textbf{Dataset Split:}
+    \begin{itemize}
+        \item The dataset is split into training and test sets.
+        \item 80\% of the data is used for training, and 20\% for testing.
+        \item Stratification ensures balanced class distribution in train/test sets.
+    \end{itemize}
+
+    \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}
+
+    \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}
+        \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}{Homogeneous GNN Model for Edge Prediction}
+    \textbf{Architecture:}
+    \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.
+        \end{itemize}
+    \end{itemize}
+\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{itemize}
+        \item \textbf{Node Types:} \texttt{stops}, \texttt{depot}
+        \item \textbf{Edge 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})
+        \end{itemize}
+        \item \textbf{Convolution Layers:} Uses \texttt{GCNConv} and \texttt{SAGEConv} for message passing.
+    \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{itemize}
+        \item Predicts sparse, directed \texttt{remember} edges using dot product of \texttt{stops} embeddings.
+    \end{itemize}
+    
+    \textbf{Goal:} Predict sparse directed edges between \texttt{stops}.
+\end{frame}
+
+%------------------------------------------------
+\section{Status}
+%------------------------------------------------
+\subsection{Results}
+\begin{frame}{Status}
+    % \begin{table}
+    %     \begin{tabular}{l l l}
+    %         \toprule
+    %         \textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2} \\
+    %         \midrule
+    %         Treatment 1         & 0.0003262           & 0.562               \\
+    %         Treatment 2         & 0.0015681           & 0.910               \\
+    %         Treatment 3         & 0.0009271           & 0.296               \\
+    %         \bottomrule
+    %     \end{tabular}
+    %     \caption{Table caption}
+    % \end{table}
+    \begin{itemize}
+        \item Over 24 000 randomly generated instances analyzed, with about 4 000 considered hard to solve, all of them of size 100 customers in the 100x100 square.
+        \item AugNg-Set available for over 20 000 of them.
+        \item Preliminar results of simple random forest allow solving most of the Solomon instances in competitive times. Models 1 and 2 are not easily distinguishable. Model 1 has default hyperparameters.
+        \item 3 models available, Model1(Acc=0,895), Model2(Acc=0,876) and Model3(Acc=0,785), all of them based on projective graph encoding using Graph Convolutional Networks (GCN).
+        \item Preliminar results of graph neural models allow solving most of the Solomon instances using model 2 in competitive times. Model 1 has the best results in terms of accuracy, but cannot propose sets for all Solomon instances. Model 3 shows lesser Accuracy but it's under development.
+    \end{itemize}
+    
+\end{frame}
+
+\begin{frame}{Random forest summary}
+    \begin{columns}[c]
+        \column{.5\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=5cm]{rf_clf1_roc.png}
+            \caption{Model1, Acc=0,923}
+        \end{figure}
+        \column{.5\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=5cm]{rf_clf2_roc.png}
+            \caption{Model2, Acc=0,920}
+        \end{figure}
+    \end{columns}
+\end{frame}
+
+\begin{frame}{Graph neural networks summary}
+    % \begin{table}
+    %     \begin{tabular}{l l l}
+    %         \toprule
+    %         \textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2} \\
+    %         \midrule
+    %         Treatment 1         & 0.0003262           & 0.562               \\
+    %         Treatment 2         & 0.0015681           & 0.910               \\
+    %         Treatment 3         & 0.0009271           & 0.296               \\
+    %         \bottomrule
+    %     \end{tabular}
+    %     \caption{Table caption}
+    % \end{table}
+    \begin{columns}[c]
+        \column{.33\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=4.5cm]{m1.png}
+            \caption{Model1, Acc=0,895}
+        \end{figure}
+        \column{.33\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=4.5cm]{m2.png}
+            \caption{Model2, Acc=0,876}
+        \end{figure}
+        \column{.33\textwidth}
+        \begin{figure}[h]
+            
+            \centering
+            \includegraphics[width=4.5cm]{m3.png}
+            \caption{Model3, Acc=0,785}
+        \end{figure}
+    \end{columns}
+    
+\end{frame}
+
+\begin{frame}{Out-of-distribution. Solomon Instances}
+    \begin{table}
+        \begin{tabular}{l l l}
+            \toprule
+            \textbf{Model} & \textbf{In-Distr Sensitivity} & \textbf{Out-Distr Sensitivity} \\
+            \midrule
+            Random Forest               & 0.930           & 0.273               \\
+            Deep Class. + GNN encoding  & 0.933           & 0.025               \\
+            Homogeneous NG encoding     & 0.064           & 0.049               \\
+            Heterogeneous NG encoding   & -           & -               \\
+            \bottomrule
+        \end{tabular}
+        \caption{Comparing sensitivity of the model with a selection of Solomon Instances.}
+    \end{table}
+\end{frame}
+\subsection{Discussion}
+\begin{frame}{Discussion}
+    \begin{itemize}
+        \item Reminder: Random Forest has access to engineered variables.
+        \item The effect of engineered variables should be studied on Deep Classification + GNN encoding on the classification head. No good results have been found on the encoding head.
+        \item Sensitivity: how to improve?
+        \item Remark: (node-) encoding a complete graph has a lot of disadvantages and some heuristics must be performed (kNN).
+        \item Heterogeneity in the nature of the graph hasn't proved interesting increases in performance. Lack of interesting properties to exploit maybe?
+        \item How to create a useful attention layer? Some research is available for adding scores to routes. In this research it's shown decreased performance (overfitting).
+    \end{itemize}
+\end{frame}
+
+%------------------------------------------------
+
+% \begin{frame}{Theorem}
+%     \begin{theorem}[Mass--energy equivalence]
+%         $E = mc^2$
+%     \end{theorem}
+% \end{frame}
+
+% %------------------------------------------------
+
+% \begin{frame}{Figure}
+%     Uncomment the code on this slide to include your own image from the same directory as the template .TeX file.
+%     %\begin{figure}
+%     %\includegraphics[width=0.8\linewidth]{test}
+%     %\end{figure}
+% \end{frame}
+
+% %------------------------------------------------
+
+% \begin{frame}[fragile] % Need to use the fragile option when verbatim is used in the slide
+%     \frametitle{Citation}
+%     An example of the \verb|\cite| command to cite within the presentation:\\~
+
+%     This statement requires citation \cite{p1}.
+% \end{frame}
+
+% %------------------------------------------------
+% \section{Planning}
+% %------------------------------------------------
+
+% \begin{frame}{Quarter projections}
+%     \includegraphics[width=0.99\linewidth]{calnov2024.jpg}
+% \end{frame}
+
+% %------------------------------------------------
+
+% \begin{frame}{Theorem}
+%     \begin{theorem}[Mass--energy equivalence]
+%         $E = mc^2$
+%     \end{theorem}
+% \end{frame}
+
+% %------------------------------------------------
+
+% \begin{frame}{Figure}
+%     Uncomment the code on this slide to include your own image from the same directory as the template .TeX file.
+%     %\begin{figure}
+%     %\includegraphics[width=0.8\linewidth]{test}
+%     %\end{figure}
+% \end{frame}
+
+% %------------------------------------------------
+
+% \begin{frame}[fragile] % Need to use the fragile option when verbatim is used in the slide
+%     \frametitle{Citation}
+%     An example of the \verb|\cite| command to cite within the presentation:\\~
+
+%     This statement requires citation \cite{p1}.
+% \end{frame}
+
+%------------------------------------------------
+
+\begin{frame}{References}
+    % Beamer does not support BibTeX so references must be inserted manually as below
+    \footnotesize{
+        \begin{thebibliography}{99}
+            \bibitem[Pecin, 2012]{p1} Pecin, Daniel (2012)
+            \newblock Branch-Cut-and-Price Algorithms for Vehicle Routing Problems
+            \newblock \emph{Column Generation 2012 Workshop}, GERAD, Montreal, Canada.
+            \newblock \url{https://www.gerad.ca/colloques/ColumnGeneration2012/presentations/session6/Pecin.pdf}.
+        \end{thebibliography}
+        \begin{thebibliography}{99}
+            \bibitem[Costa, 2019]{p3} Costa,  Luciano and Contardo,  Claudio and Desaulniers,  Guy (2019)
+            \newblock Exact Branch-Price-and-Cut Algorithms for Vehicle Routing
+            \newblock \emph{Transportation Science}
+            \newblock \url{http://dx.doi.org/10.1287/trsc.2018.0878}.
+        \end{thebibliography}
+        \begin{thebibliography}{99}
+            \bibitem[Kipf and Welling, 2017]{p2} Kipf, Thomas N., and Welling, Max (2017)
+            \newblock Semi-Supervised Classification with Graph Convolutional Networks
+            \newblock \emph{arXiv preprint}, arXiv:1609.02907.
+            \newblock \url{https://arxiv.org/abs/1609.02907}.
+        \end{thebibliography}
+
+    }
+\end{frame}
+
+%------------------------------------------------
+
+
+
+% \begin{frame}
+%     \Huge{\centerline{The End}}
+% \end{frame}
+
+\begin{frame}
+    % Print the title page as the first slide
+    \titlepage
+\end{frame}
+
+
+\section{Annex}
+
+
+%------------------------------------------------
+\begin{frame}{NG-Sets and ways of obtaining them}
+    An Ng-Path relaxation is a tool used for speeding-up elementarity in solutions obtained via SPPRC algorithm. These rely on the usage of Ng-Sets associated to each stop to compute a resource called the memory.
+    \begin{block}{Ng-Set}
+        Given $\Delta \geq 0$ a distance threshold and a graph $G=(V,E)$ with $V' = V \setminus \{0\}$ and positions $d_i $ $\forall i \in V$, the Ng-Set will be then defined by $\mathcal{N}_{i} = \{i \in V': ||d_j - d_i||_k \leq \Delta\} \forall j \neq i \in V'$
+    \end{block}
+    \begin{block}{Memory of a label}
+        Let $L$ be a label obtained in a previous iteration of an SPPRC following a path $V(L)$, then the memory associated will be $\Pi (L)=\{i_u \in V(L): i_u \in \cap_{s=u}^{k}\mathcal{N}_{i_s} \}$
+    \end{block}
+    A Label cannot be extended to a candidate that belongs to this memory.
+    
+\end{frame}
+
+%------------------------------------------------
+
+\begin{frame}{NG-Sets and ways of obtaining them}
+
+    Specifics adaptations to the SPPRC algorithms that must be defined for this memory resource are the following:
+    \begin{block}{Updating a label}
+        Let $L$ be a label obtained in a previous iteration of an SPPRC and extended to node $j$, then the new label after extension will be $\Pi (L')= \Pi (L)\cap \mathcal{N}_{j} \cup \{j\}$
+    \end{block}
+    \begin{block}{Dominance check}
+        Let $L_1, L_2$ 2 labels obtained by SPPRC. We say that $L_1$ dominates $L_2$ in sense of memory when $\Pi (L_1) \subseteq \Pi (L_2)$
+    \end{block}
+    \begin{examples}
+        There's other methods to create the Ng-Sets outside of the $\Delta$-method such as using k-NN, or iteratively reach the elementary lower bound as in the Augmented Ng-Set.
+    \end{examples}
+\end{frame}
+
+
+
+%----------------------------------------------------------------------------------------
+
+\end{document}
\ No newline at end of file