Added LP Relaxation slides
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@ -220,6 +220,550 @@ return $\boldsymbol{\hat{c}}$
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\vspace*{-1cm}
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\begin{gather*}
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\boldsymbol{G} =
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\begin{bmatrix}
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0 & 1 & 1
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\end{bmatrix} \hspace{1cm}
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\boldsymbol{H} =
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\begin{bmatrix}
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1 & 1 & 1\\
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0 & 1 & 1
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\end{bmatrix} \\[1em]
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\mathcal{C} = \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},
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\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\}
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\end{gather*}%
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\vspace*{-1cm}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.4\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.9, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\end{tikzpicture}
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\caption{Set of all codewords $\mathcal{C}$}
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\label{fig:lp:poly:exact_ilp}
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\end{subfigure}%
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\begin{subfigure}[c]{0.16\textwidth}
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\centering
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\begin{tikzpicture}
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\node (relaxation) at (0, 0) {Relaxation};
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\draw (-1.5, 0) -- (relaxation);
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\draw[->] (relaxation) -- (1.5, 0);
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.4\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.9, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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% Polytope Edges
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\end{tikzpicture}
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\caption{Codeword polytope $\text{poly}\left( \mathcal{C} \right) $}
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\label{fig:lp:poly:exact}
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\end{subfigure}
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\end{figure}
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\end{frame}
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\vspace*{-8mm}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{minipage}{\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.7, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c101) at (p101) {};
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\node[codeword] (c110) at (p110) {};
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\node[codeword] (c011) at (p011) {};
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% Polytope Edges & Faces
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\draw[line width=1pt, color=KITblue] (c000) -- (c101);
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\draw[line width=1pt, color=KITblue] (c000) -- (c110);
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITblue] (c101) -- (c110);
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\draw[line width=1pt, color=KITblue] (c101) -- (c011);
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\draw[line width=1pt, color=KITblue] (c011) -- (c110);
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\fill[KITblue, opacity=0.15] (p000) -- (p101) -- (p011) -- cycle;
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\fill[KITblue, opacity=0.15] (p000) -- (p110) -- (p101) -- cycle;
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\fill[KITblue, opacity=0.15] (p110) -- (p011) -- (p101) -- cycle;
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, right=0.07cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, align=center] at (-4,1)
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{$j=1$\\ $\left( c_1 + c_2+ c_3 = 0 \right) $};
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\end{tikzpicture}
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\end{minipage}
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\vspace{5mm}
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\begin{minipage}{\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.7, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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\node[codeword] (c100) at (p100) {};
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\node[codeword] (c111) at (p111) {};
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% Polytope Edges & Faces
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITblue] (c000) -- (c100);
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\draw[line width=1pt, color=KITblue] (c100) -- (c111);
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\draw[line width=1pt, color=KITblue] (c111) -- (c011);
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\fill[KITblue, opacity=0.2] (p000) -- (p100) -- (p111) -- (p011) -- cycle;
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
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\node[color=KITblue, align=center] at (-4,1)
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{$j=2$\\ $\left(c_2 + c_3 = 0\right)$};
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\end{tikzpicture}
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\end{minipage}
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\caption{Local codeword polytopes of the check nodes}
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\label{fig:lp:poly:local}
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\end{subfigure}%
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\begin{subfigure}[c]{0.18\textwidth}
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\centering
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\begin{tikzpicture}
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\draw[densely dashed] (0, -2) -- (0, 2);
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\draw[densely dashed] (-0.5, -2) -- (0, -2);
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\draw[densely dashed] (-0.5, 2) -- (0, 2);
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\node (intersection) at (1.5, 0) {Intersection};
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\draw[densely dashed] (0, 0) -- (intersection);
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\draw[densely dashed, ->] (intersection) -- (3,0);
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.32\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.9, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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\node[pseudocodeword] (cpseudo) at (2, 1, 1) {};
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% Polytope Edges & Faces
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITred] (cpseudo) -- (c000);
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\draw[line width=1pt, color=KITred] (cpseudo) -- (c011);
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\fill[KITred, opacity=0.2] (p000) -- (p011) -- (2,1,1) -- cycle;
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITred, right=0cm of cpseudo]
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{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
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\end{tikzpicture}
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\caption{Relaxed codeword polytope $\overline{Q}$}
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\label{fig:lp:poly:relaxed}
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\end{subfigure}
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\end{figure}
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\end{frame}
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.2\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.8, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node at (3, -2) {$\mathcal{C}$};
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.18\textwidth}
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\centering
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\begin{tikzpicture}
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\node (relaxation) at (0, 0) {Relaxation};
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\draw (-1.5, 0) -- (relaxation);
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\draw[->] (relaxation) -- (1.5, 0);
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.2\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
|
||||
draw, circle, inner sep=0pt, minimum size=4pt]
|
||||
\tdplotsetmaincoords{60}{25}
|
||||
\begin{tikzpicture}[scale=0.8, tdplot_main_coords]
|
||||
% Cube
|
||||
|
||||
\coordinate (p000) at (0, 0, 0);
|
||||
\coordinate (p001) at (0, 0, 2);
|
||||
\coordinate (p010) at (0, 2, 0);
|
||||
\coordinate (p011) at (0, 2, 2);
|
||||
\coordinate (p100) at (2, 0, 0);
|
||||
\coordinate (p101) at (2, 0, 2);
|
||||
\coordinate (p110) at (2, 2, 0);
|
||||
\coordinate (p111) at (2, 2, 2);
|
||||
|
||||
\draw[] (p000) -- (p100);
|
||||
\draw[] (p100) -- (p101);
|
||||
\draw[] (p101) -- (p001);
|
||||
\draw[] (p001) -- (p000);
|
||||
|
||||
\draw[dashed] (p010) -- (p110);
|
||||
\draw[] (p110) -- (p111);
|
||||
\draw[] (p111) -- (p011);
|
||||
\draw[dashed] (p011) -- (p010);
|
||||
|
||||
\draw[dashed] (p000) -- (p010);
|
||||
\draw[] (p100) -- (p110);
|
||||
\draw[] (p101) -- (p111);
|
||||
\draw[] (p001) -- (p011);
|
||||
|
||||
% Polytope Vertices
|
||||
|
||||
\node[codeword] (c000) at (p000) {};
|
||||
\node[codeword] (c011) at (p011) {};
|
||||
|
||||
% Polytope Edges
|
||||
|
||||
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
|
||||
|
||||
% Polytope Annotations
|
||||
|
||||
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
|
||||
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
|
||||
|
||||
\node at (3, -2) {$\text{poly}\left(\mathcal{C}\right)$};
|
||||
\end{tikzpicture}
|
||||
\end{subfigure}%
|
||||
\begin{subfigure}[c]{0.18\textwidth}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\node (relaxation) at (0, 0) {Relaxation};
|
||||
|
||||
\draw (-1.5, 0) -- (relaxation);
|
||||
\draw[->] (relaxation) -- (1.5, 0);
|
||||
\end{tikzpicture}
|
||||
\end{subfigure}%
|
||||
\begin{subfigure}[c]{0.2\textwidth}
|
||||
\centering
|
||||
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
|
||||
draw, circle, inner sep=0pt, minimum size=4pt]
|
||||
\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
|
||||
draw, circle, inner sep=0pt, minimum size=4pt]
|
||||
\tdplotsetmaincoords{60}{25}
|
||||
\begin{tikzpicture}[scale=0.8, tdplot_main_coords]
|
||||
% Cube
|
||||
|
||||
\coordinate (p000) at (0, 0, 0);
|
||||
\coordinate (p001) at (0, 0, 2);
|
||||
\coordinate (p010) at (0, 2, 0);
|
||||
\coordinate (p011) at (0, 2, 2);
|
||||
\coordinate (p100) at (2, 0, 0);
|
||||
\coordinate (p101) at (2, 0, 2);
|
||||
\coordinate (p110) at (2, 2, 0);
|
||||
\coordinate (p111) at (2, 2, 2);
|
||||
|
||||
\draw[] (p000) -- (p100);
|
||||
\draw[] (p100) -- (p101);
|
||||
\draw[] (p101) -- (p001);
|
||||
\draw[] (p001) -- (p000);
|
||||
|
||||
\draw[dashed] (p010) -- (p110);
|
||||
\draw[] (p110) -- (p111);
|
||||
\draw[] (p111) -- (p011);
|
||||
\draw[dashed] (p011) -- (p010);
|
||||
|
||||
\draw[dashed] (p000) -- (p010);
|
||||
\draw[] (p100) -- (p110);
|
||||
\draw[] (p101) -- (p111);
|
||||
\draw[] (p001) -- (p011);
|
||||
|
||||
% Polytope Vertices
|
||||
|
||||
\node[codeword] (c000) at (p000) {};
|
||||
\node[codeword] (c011) at (p011) {};
|
||||
\node[pseudocodeword] (cpseudo) at (2, 1, 1) {};
|
||||
|
||||
% Polytope Edges & Faces
|
||||
|
||||
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
|
||||
\draw[line width=1pt, color=KITred] (cpseudo) -- (c000);
|
||||
\draw[line width=1pt, color=KITred] (cpseudo) -- (c011);
|
||||
|
||||
\fill[KITred, opacity=0.2] (p000) -- (p011) -- (2,1,1) -- cycle;
|
||||
|
||||
% Polytope Annotations
|
||||
|
||||
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
|
||||
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
|
||||
\node[color=KITred, right=0cm of cpseudo]
|
||||
{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
|
||||
|
||||
\node at (3, -2) {$\overline{Q}$};
|
||||
\end{tikzpicture}
|
||||
\end{subfigure}%
|
||||
|
||||
|
||||
\end{figure}
|
||||
|
||||
\end{frame}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\begin{frame}[t]
|
||||
% \frametitle{LP Relaxation}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user