Commit 9c21de91 authored by Michael Keller's avatar Michael Keller
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Thesis: NP Stuff

parent 001b6d80
......@@ -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}
\ No newline at end of file
\chapter{Gradient Descent}
\ No newline at end of file
......@@ -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
\chapter{Helpful Statements}
\section{Growing solutions}
\section{Creating integer solutions from fractional solutions}
\ No newline at end of file
\chapter{The Linear Programming Method}
\ No newline at end of file
......@@ -6,6 +6,11 @@
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......@@ -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
......
......@@ -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}%
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