Commit effde74e authored by Michael Keller's avatar Michael Keller
Browse files

Integral Advanced done

parent e9d1e9cd
......@@ -160,7 +160,11 @@ statement with a condition on $R$ might be helpful:
\end{figure}
\FloatBarrier
The interesting score change is the following quadratic one
between the pixels $u$ and $v$:
between the pixels $u$ and $v$. Here $p_c$ denotes
the amount of crop $c$ pixel $p$ contains before
the swap. $p_c'$ is defined analogously for after
the swap.
\begin{align*}
&\sum_{c_i \in \Cps} \sum_{c_j \in \Cps} R(c_i, c_j) \cdot (u'_i \cdot v'_{j} - u_i \cdot v_{j})\\
&= \sum_{c_i \in \{c_\alpha, c_\beta \}}
......@@ -222,9 +226,25 @@ statement with a condition on $R$ might be helpful:
&+ \lambda \cdot (R(c_\beta, c_\alpha) - R(c_\alpha, c_\alpha)) \cdot v_\alpha\\
&+ \lambda \cdot (R(c_\beta, c_\beta) - R(c_\alpha, c_\beta)) \cdot v_\beta
\end{align*}
Which, if every crop likes all other crops
more than itself, can be made positive with the
appropriate $\lambda$. Then the quadratic
term in the formula above will always be positive,
and we can focus only on all of the linear constraints.
Of special note here is the quadratic term.
If every crop likes all other crops
at least as much as it likes itself,
then the quadratic term is positive. The quadratic
term being positive means we can always swap
in at least one direction without
the score getting worse. The optimal swap
direction is determined by all of the linear
functions in $\lambda$ we put to the side
throughout this proof. Therefore
a relationship function $R$ fulfilling
$R(c_i, c_j) \geq R(c_i, c_i)$ ensures there is
always a good way to swap crops among
neighboring pixels.
Being allowed to swap even amongst neighboring
pixels allows us to disregard the pattern
structure we had for the standard swapping method
and we can by the same reasoning conclude
that this new method has after termination
at most $\binom{C}{2}$ still fractional pixels.
\end{proof}
\ No newline at end of file
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......@@ -4,7 +4,7 @@
\contentsline {proof}{{Proof}{2}{}}{19}{proof.2}%
\contentsline {theorem}{{Theorem}{7.{2}}{}}{20}{theorem.7.2.2}%
\contentsline {proof}{{Proof}{3}{}}{21}{proof.3}%
\contentsline {theorem}{{Theorem}{8.{1}}{}}{25}{theorem.8.1.1}%
\contentsline {proof}{{Proof}{4}{}}{25}{proof.4}%
\contentsline {theorem}{{Theorem}{8.{2}}{}}{25}{theorem.8.1.2}%
\contentsline {proof}{{Proof}{5}{}}{25}{proof.5}%
\contentsline {theorem}{{Theorem}{8.{1}}{}}{27}{theorem.8.1.1}%
\contentsline {proof}{{Proof}{4}{}}{27}{proof.4}%
\contentsline {theorem}{{Theorem}{8.{2}}{}}{27}{theorem.8.1.2}%
\contentsline {proof}{{Proof}{5}{}}{27}{proof.5}%
......@@ -19,10 +19,10 @@
\contentsline {section}{\numberline {7.2}Creating integer solutions from fractional solutions}{19}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}The standard method}{19}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}The advanced method}{20}{subsection.7.2.2}%
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{23}{chapter.8}%
\contentsline {section}{\numberline {8.1}Problem Setup}{23}{section.8.1}%
\contentsline {section}{\numberline {8.2}Method}{26}{section.8.2}%
\contentsline {chapter}{\chapternumberline {9}Gradient Descent}{29}{chapter.9}%
\contentsline {chapter}{\chapternumberline {10}Conclusion}{31}{chapter.10}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{33}{appendix.A}%
\contentsline {chapter}{Bibliography}{35}{appendix*.4}%
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{25}{chapter.8}%
\contentsline {section}{\numberline {8.1}Problem Setup}{25}{section.8.1}%
\contentsline {section}{\numberline {8.2}Method}{28}{section.8.2}%
\contentsline {chapter}{\chapternumberline {9}Gradient Descent}{31}{chapter.9}%
\contentsline {chapter}{\chapternumberline {10}Conclusion}{33}{chapter.10}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{35}{appendix.A}%
\contentsline {chapter}{Bibliography}{37}{appendix*.4}%
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