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\changetocdepth {2}
\babel@toc {english}{}
\contentsline {chapter}{Contents}{iii}{section*.1}%
\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}%
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\contentsline {section}{\numberline {3.1}Integer Version}{5}{section.3.1}%
\contentsline {section}{\numberline {3.2}Fractional Version}{6}{section.3.2}%
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\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}%
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\contentsline {section}{\numberline {4.3}Fractional Pixel Farming is NP Complete}{9}{section.4.3}%
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\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}%
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\contentsline {section}{\numberline {6.3}LP bound}{17}{section.6.3}%
\contentsline {chapter}{\chapternumberline {7}Helpful Statements}{19}{chapter.7}%
\contentsline {section}{\numberline {7.1}Growing solutions}{19}{section.7.1}%
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\contentsline {section}{\numberline {7.2}Creating integer solutions from fractional solutions}{21}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}The standard method}{21}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}The advanced method}{22}{subsection.7.2.2}%
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{27}{chapter.8}%
\contentsline {section}{\numberline {8.1}Problem Setup}{27}{section.8.1}%
\contentsline {section}{\numberline {8.2}Method}{30}{section.8.2}%
\contentsline {chapter}{\chapternumberline {9}Gradient Descent}{33}{chapter.9}%
\contentsline {chapter}{\chapternumberline {10}Conclusion}{35}{chapter.10}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{37}{appendix.A}%
\contentsline {chapter}{Bibliography}{39}{appendix*.4}%