Thesis: NP Stuff

Thesis/extrapackages.tex

@@ -59,3 +59,9 @@
 
 %% [REC] Nicer tables.  Read the excellent documentation.
 \usepackage{booktabs}
+
+%% [OPT] Added by kelmich. Allows drawing of graphs + other fancy things
+\usepackage{tikz}
+
+%% [OPT] Added by kelmich. Allows for text immediately after figures.
+\usepackage{placeins}
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Thesis/gradient-descent.tex

+\chapter{Gradient Descent}
\ No newline at end of file

Thesis/hardness.tex

@@ -54,8 +54,75 @@ and $C$ for the pixel farming decision problem. We are asked
 to solve the hamiltonian path decision problem in polynomial
 time in relation to the size of $G$.
 
-In polynomial time we build the following pixel farming
-decision problem. Our crops are as follows:
-\begin{displaymath}
-    \Cps = V \cup \{ \abs{V} + 1, \dots, \abs{V} + 1 + l +  \}
-\end{displaymath}
\ No newline at end of file
+Let us first look at a simple case, when $l = \abs{V}$.
+The core idea is that we translate vertices to crops and
+edges to crops liking each other. Then, a pixel farming
+decision problem with a score that means every pixel
+likes both of its neighbors must imply that there exists
+a hamiltonian path in the graph $G$. Formally we build
+the following pixel farming decision problem:
+\begin{align*}
+    X &= \abs{V} && Y = 1\\
+    \Cps &= V && R(a, b) = \begin{cases}
+        1 & \{a, b\} \in E\\
+        0 & otherwise.
+    \end{cases}\\
+    s &= 2 \cdot (\abs{V} - 1) && D(c) = \frac{1}{\abs{V}} \quad \forall c \in \Cps
+\end{align*}
+The following example illustrates this:
+\begin{figure}[h]
+    \centering
+    \begin{minipage}{0.45\textwidth}
+        \centering
+        \begin{tikzpicture}[node distance={15mm}, main/.style = {draw, circle}]
+            \node[main] (1) {$v_1$}; 
+            \node[main] (2) [right of=1] {$v_2$};
+            \node[main] (3) [below of=2] {$v_3$}; 
+            \node[main] (4) [below of=1] {$v_4$};
+            \draw (1) -- (2);
+            \draw (2) -- (4);
+            \draw (3) -- (1);
+            \draw (4) -- (3);
+        \end{tikzpicture} 
+        \caption{example graph}
+    \end{minipage}\hfill
+    \begin{minipage}{0.45\textwidth}
+        \centering
+        \begin{tikzpicture}[node distance={0.75cm}, main/.style = {draw, minimum size=0.75cm}]
+            \node[main] (1) {$v_1$};
+            \node[main] (2) [right of=1] {$v_2$};
+            \node[main] (3) [right of=2] {$v_4$}; 
+            \node[main] (4) [right of=3] {$v_3$};
+        \end{tikzpicture}
+        \hfill
+        \caption{corresponding solution field}
+    \end{minipage}
+\end{figure}
+\FloatBarrier
+What about the case where $l < \abs{V}$? For this
+we need to resort to a trick. Say for example we are
+looking for a path of length at least $\abs{V} - 2$.
+We can't simply change $s$ to be $4$ less than before,
+because we have no guarantees as to where these ``cuts''
+will be. We could end up in a situation where our
+pixel farming algorithm gives us 3 paths that are all
+not of sufficient length.
+\begin{figure}[h]
+    \centering
+    \begin{tikzpicture}[node distance={0.5cm}, main/.style = {draw, minimum size=0.5cm}]
+        \node[main] (1) {};
+        \node[main] (13) [below of =1] {};
+        \foreach \x in {2,...,12}
+            {   
+                \pgfmathtruncatemacro{\prev}{\x - 1}
+                \pgfmathtruncatemacro{\below}{\x + 11}
+                \node[main] (\x) [right of=\prev] {};
+                \node[main] (\below) [below of=\x] {};
+            }
+            \node[main] (26) [below of=20] {};
+            \node[main] (27) [below of=22] {};
+    \end{tikzpicture}
+    \hfill
+    \caption{example structure}
+\end{figure}
+\FloatBarrier
\ No newline at end of file

Thesis/helpers.tex

+\chapter{Helpful Statements}
+
+\section{Growing solutions}
+
+\section{Creating integer solutions from fractional solutions}
\ No newline at end of file

Thesis/lp-method.tex

+\chapter{The Linear Programming Method}
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Thesis/thesis.out

@@ -6,6 +6,11 @@
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Thesis/thesis.tex

@@ -116,6 +116,9 @@
 \input{related-work.tex}
 \input{problem-statement.tex}
 \input{hardness.tex}
+\input{helpers.tex}
+\input{lp-method.tex}
+\input{gradient-descent.tex}
 \input{conclusion.tex}
 
 \appendix

Thesis/thesis.toc

@@ -10,6 +10,11 @@
 \contentsline {section}{\numberline {4.3}Pixel Farming finds Hamiltonian Paths}{8}{section.4.3}%
 \contentsline {paragraph}{Hamiltonian Paths}{8}{section*.2}%
 \contentsline {paragraph}{Reduction}{8}{section*.3}%
-\contentsline {chapter}{\chapternumberline {5}Conclusion}{9}{chapter.5}%
-\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{11}{appendix.A}%
-\contentsline {chapter}{Bibliography}{13}{appendix*.4}%
+\contentsline {chapter}{\chapternumberline {5}Helpful Statements}{11}{chapter.5}%
+\contentsline {section}{\numberline {5.1}Growing solutions}{11}{section.5.1}%
+\contentsline {section}{\numberline {5.2}Creating integer solutions from fractional solutions}{11}{section.5.2}%
+\contentsline {chapter}{\chapternumberline {6}The Linear Programming Method}{13}{chapter.6}%
+\contentsline {chapter}{\chapternumberline {7}Gradient Descent}{15}{chapter.7}%
+\contentsline {chapter}{\chapternumberline {8}Conclusion}{17}{chapter.8}%
+\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{19}{appendix.A}%
+\contentsline {chapter}{Bibliography}{21}{appendix*.4}%