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zFfd|eH8VG6WnyAsH8n9cVq-QrVP#=4HaR{#Vl-qjH8*28FgG?ZHe+RDHe+ElFk&+| zGGR41HZeCjK3yO_J_>Vma%Ev{3V57jVgLaqMrIKY1|}e@1Vl13f=I>3}h-CW1002Ce50aBT0&15q_yiUIpuq9~ diff --git a/Report/Graver_Basis_tex/Thesis.tex b/Report/Graver_Basis_tex/Thesis.tex index 14767f3..a880da4 100644 --- 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 new file mode 100644 index 0000000..0f55c02 --- /dev/null +++ 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 + +