Commit 6f378044 authored by Michael Keller's avatar Michael Keller
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Thesis: Added Tables for R matrices

parent 98b3f275
\chapter{Benchmark Problems}
\ No newline at end of file
\chapter{Benchmark Problems}
We received two $R$ matrices from Agroscope. However,
the second matrix is the more interesting one for two
reasons:
\begin{itemize}
\item It turns out to be harder to optimize
\item Agroscope plans on conducting a field experiment with
the second matrix
\end{itemize}
The first matrix is:
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{ c c c c c c c c c c c c }
& $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$ & $c_{11}$ \\
$c_1$ & $-0.5$ & $1$ & $1$ & $1$ & $1$ & $1$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $1$ \\
$c_2$ & $1$ & $-0.5$ & $-1$ & $1$ & $1$ & $0.5$ & $0.5$ & $-1$ & $1$ & $1$ & $1$ \\
$c_3$ & $1$ & $-1$ & $-0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $0$ & $0$ & $0$ \\
$c_4$ & $1$ & $1$ & $-1$ & $-0.5$ & $0$ & $0$ & $0.5$ & $-1$ & $1$ & $-1$ & $0$ \\
$c_5$ & $1$ & $1$ & $1$ & $0$ & $-0.5$ & $0$ & $0.5$ & $1$ & $1$ & $1$ & $1$ \\
$c_6$ & $1$ & $0.5$ & $0$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $0$ & $0$ \\
$c_7$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ & $0.5$ \\
$c_8$ & $-1$ & $-1$ & $0.5$ & $-1$ & $1$ & $0$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $0$ \\
$c_9$ & $0.5$ & $1$ & $0$ & $1$ & $1$ & $0$ & $0.5$ & $0$ & $-0.5$ & $1$ & $0$ \\
$c_{10}$ & $0.5$ & $1$ & $0$ & $-1$ & $1$ & $0$ & $0.5$ & $0.5$ & $1$ & $-0.5$ & $1$\\
$c_{11}$ & $1$ & $1$ & $0$ & $0$ & $1$ & $0$ & $0.5$ & $0$ & $0$ & $1$ & $-0.5$ \\
\end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}llr@{}} \toprule
Crop Id & Crop Name\\ \midrule
$c_1$ & Kreuzblütler \\
$c_2$ & Schmetterlingsblütler \\
$c_3$ & Nachtschattengewächse \\
$c_4$ & Kartoffel \\
$c_5$ & Korbblütler \\
$c_6$ & Getreide \\
$c_7$ & Kräuter \\
$c_8$ & Liliengewächse \\
$c_9$ & Süssgräser \\
$c_{10}$ & Kürbisgewächse \\
$c_{11}$ & Gänsefussgewächse \\ \bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
and the second matrix is:
\FloatBarrier
\begin{figure}
\centering
\begin{tabular}{ c c c c c c c c c c c }
x & $c_1$ & $c_2$ & $c_3$ & $c_4$ & $c_5$ & $c_6$ & $c_7$ & $c_8$ & $c_9$ & $c_{10}$\\
$c_1$ & $-1$ & $0$ & $0.5$ & $-0.5$ & $1$ & $0$ & $0.5$ & $1$ & $0.5$ & $0$\\
$c_2$ & $0$ & $-1$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $0.5$ & $0$\\
$c_3$ & $0.5$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0.5$ & $0.5$ & $0$ & $-0.5$\\
$c_4$ & $-0.5$ & $1$ & $0$ & $-1$ & $0$ & $0$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
$c_5$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $0$ & $0$ & $0$ & $0$ & $0$\\
$c_6$ & $0$ & $0$ & $0$ & $0$ & $0$ & $-1$ & $1$ & $0.5$ & $0$ & $0$\\
$c_7$ & $0.5$ & $0.5$ & $0.5$ & $-0.5$ & $0$ & $0.5$ & $-1$ & $0.5$ & $0.5$ & $0.5$\\
$c_8$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\
$c_9$ & $0.5$ & $0.5$ & $0$ & $1$ & $0$ & $0$ & $0.5$ & $0.5$ & $-1$ & $0$\\
$c_{10}$ & $0$ & $0$ & $-0.5$ & $0.5$ & $0$ & $0$ & $1$ & $0.5$ & $0.5$ & $-1$\\
\end{tabular}
\end{figure}
\FloatBarrier
where
\FloatBarrier
\begin{figure}[h]
\centering
\begin{tabular}{@{}llr@{}} \toprule
Crop Id & Crop Name\\ \midrule
$c_1$ & Weisskohl \\
$c_2$ & Karotten \\
$c_3$ & Kartoffeln \\
$c_4$ & Zwiebeln \\
$c_5$ & Hanf \\
$c_6$ & Zuckermais \\
$c_7$ & Buschbohnen \\
$c_8$ & Kräutermischung \\
$c_9$ & Kopfsalat \\
$c_{10}$ & Randen \\ \bottomrule
\end{tabular}
\end{figure}
\FloatBarrier
\ No newline at end of file
\chapter{Bounds}
\section{Basic Bound}
\section{Gärtner Bound}
\ No newline at end of file
......@@ -109,7 +109,7 @@ There might be situations where up to $4 \cdot \binom{C}{2}$
still fractional pixels are unacceptable. This might be the
case in relatively small fields with relatively large amounts
of different crops. Here the following slightly stronger
statement with a weak condition on $R$ might be helpful:
statement with a condition on $R$ might be helpful:
\begin{theorem}
If every crop prefers all other crops over itself,
......@@ -125,4 +125,9 @@ statement with a weak condition on $R$ might be helpful:
as with the previous theorem. The change we make is
that we allow swapping of crops between neighboring
pixels.
We now need to know what condition needs to be
met to ensure that swapping in either direction
even among neighboring pixels is never detrimental.
\end{proof}
\ No newline at end of file
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......@@ -115,8 +115,9 @@
\input{indroduction.tex}
\input{related-work.tex}
\input{problem-statement.tex}
\input{benchmarks.tex}
\input{hardness.tex}
\input{benchmarks.tex}
\input{bounds.tex}
\input{helpers.tex}
\input{lp-method.tex}
\input{gradient-descent.tex}
......
\contentsline {theorem}{{Theorem}{5.{1}}{}}{9}{theorem.5.2.1}%
\contentsline {proof}{{Proof}{1}{}}{9}{proof.1}%
\contentsline {theorem}{{Theorem}{6.{1}}{}}{13}{theorem.6.2.1}%
\contentsline {proof}{{Proof}{2}{}}{13}{proof.2}%
\contentsline {theorem}{{Theorem}{6.{2}}{}}{14}{theorem.6.2.2}%
\contentsline {proof}{{Proof}{3}{}}{15}{proof.3}%
\contentsline {theorem}{{Theorem}{4.{1}}{}}{7}{theorem.4.2.1}%
\contentsline {proof}{{Proof}{1}{}}{7}{proof.1}%
\contentsline {theorem}{{Theorem}{7.{1}}{}}{17}{theorem.7.2.1}%
\contentsline {proof}{{Proof}{2}{}}{17}{proof.2}%
\contentsline {theorem}{{Theorem}{7.{2}}{}}{18}{theorem.7.2.2}%
\contentsline {proof}{{Proof}{3}{}}{19}{proof.3}%
......@@ -4,19 +4,22 @@
\contentsline {chapter}{\chapternumberline {1}Introduction}{1}{chapter.1}%
\contentsline {chapter}{\chapternumberline {2}Related Work}{3}{chapter.2}%
\contentsline {chapter}{\chapternumberline {3}Problem Statement}{5}{chapter.3}%
\contentsline {chapter}{\chapternumberline {4}Benchmark Problems}{7}{chapter.4}%
\contentsline {chapter}{\chapternumberline {5}Computational Hardness}{9}{chapter.5}%
\contentsline {section}{\numberline {5.1}The decision version of pixel farming}{9}{section.5.1}%
\contentsline {section}{\numberline {5.2}Pixel Farming is NP complete}{9}{section.5.2}%
\contentsline {paragraph}{Hamiltonian Paths}{10}{section*.2}%
\contentsline {paragraph}{Reduction}{10}{section*.3}%
\contentsline {chapter}{\chapternumberline {6}Helpful Statements}{13}{chapter.6}%
\contentsline {section}{\numberline {6.1}Growing solutions}{13}{section.6.1}%
\contentsline {section}{\numberline {6.2}Creating integer solutions from fractional solutions}{13}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}The standard method}{13}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}The advanced method}{14}{subsection.6.2.2}%
\contentsline {chapter}{\chapternumberline {7}The Linear Programming Method}{17}{chapter.7}%
\contentsline {chapter}{\chapternumberline {8}Gradient Descent}{19}{chapter.8}%
\contentsline {chapter}{\chapternumberline {9}Conclusion}{21}{chapter.9}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{23}{appendix.A}%
\contentsline {chapter}{Bibliography}{25}{appendix*.4}%
\contentsline {chapter}{\chapternumberline {4}Computational Hardness}{7}{chapter.4}%
\contentsline {section}{\numberline {4.1}The decision version of pixel farming}{7}{section.4.1}%
\contentsline {section}{\numberline {4.2}Pixel Farming is NP complete}{7}{section.4.2}%
\contentsline {paragraph}{Hamiltonian Paths}{8}{section*.2}%
\contentsline {paragraph}{Reduction}{8}{section*.3}%
\contentsline {chapter}{\chapternumberline {5}Benchmark Problems}{11}{chapter.5}%
\contentsline {chapter}{\chapternumberline {6}Bounds}{15}{chapter.6}%
\contentsline {section}{\numberline {6.1}Basic Bound}{15}{section.6.1}%
\contentsline {section}{\numberline {6.2}Gärtner Bound}{15}{section.6.2}%
\contentsline {chapter}{\chapternumberline {7}Helpful Statements}{17}{chapter.7}%
\contentsline {section}{\numberline {7.1}Growing solutions}{17}{section.7.1}%
\contentsline {section}{\numberline {7.2}Creating integer solutions from fractional solutions}{17}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}The standard method}{17}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}The advanced method}{18}{subsection.7.2.2}%
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{21}{chapter.8}%
\contentsline {chapter}{\chapternumberline {9}Gradient Descent}{23}{chapter.9}%
\contentsline {chapter}{\chapternumberline {10}Conclusion}{25}{chapter.10}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{27}{appendix.A}%
\contentsline {chapter}{Bibliography}{29}{appendix*.4}%
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