Report N-Fold IP introduction

frblazquez · Jan 8, 2021 · a1c9c6f · a1c9c6f
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diff --git a/Report/Graver_Basis.pdf b/Report/Graver_Basis.pdf
diff --git a/Report/Graver_Basis_tex/Thesis.tex b/Report/Graver_Basis_tex/Thesis.tex
@@ -275,7 +275,7 @@
 
 \input{src/2.GraverBasis} % Graver Basis definition and properties
 
-%\input{src/3.NFold} % Fundamentals
+\input{src/3.NFold} % Fundamentals
 
 %\input{src/4.NFoldAugmentation} % N-Fold augmentation algorithm
 

diff --git a/Report/Graver_Basis_tex/src/3.NFold.tex b/Report/Graver_Basis_tex/src/3.NFold.tex
@@ -0,0 +1,24 @@
+\chapter{N-Fold} \label{methodology}
+
+% De Loera, Hemmecke, Onn and Weismantel 2006 paper
+
+
+A generalized N-Fold IP has constriction matrix A of the form ($A_i \in \mathbb{Z}^{rxt}, B_i \in \mathbb{Z}^{sxt}$):\\
+\begin{equation*}
+N = 
+\begin{pmatrix}
+A_1 & A_2 & \cdots & A_n \\
+B_1 & 0   & \cdots & 0 \\
+0   & B_2 & \cdots & 0 \\
+\vdots    & \vdots & \ddots & \vdots  \\
+0   & 0   & \cdots & B_n 
+\end{pmatrix}
+\end{equation*}
+
+% TODO - ADD:
+% Presentation of the N-Fold IP (generalized or not??)
+% Particularities
+% Steinitz Lemma? Or beter in next section?
+% Theoretical justification of N-Fold being polynomially solvable
+
+