Commit bda93dbd authored by Michael Keller's avatar Michael Keller
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Defined Fractional Problem

parent effde74e
......@@ -143,4 +143,54 @@ hence no path of length $\abs{V} - 1$ can exist.
As we now know that pixel farming is in NP and NP-hard,
we can conclude that pixel farming must be NP-complete.
\section{Fractional Pixel Farming is NP Complete}
We consider a slightly modified version of the
pixel farming decision problem. Instead of allowing
a pixel to only contain a single crop,
a pixel may now contain multiple crops. We do however
still require that the total amount of crops planted
within a pixel sums to one. One might suspect that
such a relaxation might enable us to formulate a
convex optimization problem. We show however that
this relaxation is also NP complete.
\begin{theorem}
The fractional pixel farming problem (decision version) is NP complete.
\end{theorem}
\begin{proof}
We want to show that the question: ``Does there exist a
fractional solution to the pixel farming problem
such that the score $S(F, R)$ for a field $F$ and
relationship function $R$ is at least $s$?'' is
NP complete.
The problem is obviously in NP, as we can calculate the
score of a given field $F$ and relationship function
$R$ in polynomial time.
For the proof of NP Hardness we again reduce to the
hamiltonian paths problem as in the proof that
regular pixel farming is NP complete. We show that
the following implications hold for any graph $G$:
\begin{enumerate}
\item There exists no optimal fractional solution to
the constructed pixel farming problem $\implies$
There exists no Hamiltonian path in $G$
\item There exists an optimal fractional
solution to the constructed pixel farming problem
$\implies$ There exists a Hamiltonian path in $G$
\end{enumerate}
\end{proof}
The first implication is trivial.
The second implication we prove indirectly. We show instead
that the following statement holds:
``There exists no Hamiltonian path in $G$''
$\implies$ ``No optimal fractional
solution to the constructed pixel farming problem exists.''
\end{proof}
\ No newline at end of file
\chapter{Problem Statement}
\section{Integer Version}
We assume that we are given four inputs:
\begin{enumerate}
\item The field dimensions, denoted as $X$ for the number of pixels
......@@ -19,18 +21,47 @@ We assume that we are given four inputs:
Further let us define a neighborhood function that gives
us all neighboring pixels of a given pixel:
\begin{displaymath}
N(x, y) = \{(a, b) \in \N^2 \ | \ a < X \land b < Y \land \abs{a - x} \leq 1 \land \abs{b - y} \leq 1 \land (a, b) \neq (x, y) \}
N((x, y)) = \{(a, b) \in \N^2 \ | \ a < X \land b < Y \land \abs{a - x} \leq 1 \land \abs{b - y} \leq 1 \land (a, b) \neq (x, y) \}
\end{displaymath}
and we are looking for a matrix $F \in \Cps^{X \times Y}$
that maximizes the following score function:
\begin{displaymath}
S(F, R) = \sum_{i = 0}^{X-1} \sum_{j = 0}^{Y-1} \sum_{n \in N(i, j)} R(F_{(i, j)}, F_n)
S(F, R) = \sum_{i = 0}^{X-1} \sum_{j = 0}^{Y-1} \sum_{n \in N((i, j))} R(F_{(i, j)}, F_n)
\end{displaymath}
where we also fulfill the distribution constraint:
\begin{align*}
I((i, j), c) &= \begin{cases}
1 & F((i, j)) = c\\
1 & F_{(i, j)} = c\\
0 & otherwise.
\end{cases}\\
\forall c \in \Cps : D(c) \cdot X \cdot Y &= \sum_{i = 0}^{X-1} \sum_{j = 0}^{Y-1} I((i, j), c)
\end{align*}
\section{Fractional Version}
The fractional version of pixel farming
we define analogously to the integer version
with the following exception: Instead of searching
for a Field $F \in \Cps^{X \times Y}$, we search for
an $F' \in \R_+^{C \times X \times Y}$. The idea
is that every pixel can contain not only a single
crop, but any combination of crops. To this end
we introduce $C$ variables for every pixel to denote
the amount of each crop a pixel contains. For example
a pixel could contain half of an carrot, a quarter
of an onion and quarter of broccoli.
We stress that we are interested in this version
of the pixel farming problem not for it's practical
applications, but rather because it enables us to
explore more traditional methods for exploring
large search spaces.
We define an analogous score function:
\begin{align*}
S(F', R) &= \sum_{i = 0}^{X-1} \sum_{j = 0}^{Y-1} \sum_{n \in N((i, j))} \sum_{c_i \in \Cps} \sum_{c_j \in \Cps} R(c_i, c_j) \cdot F_{(i, j)}^{c_i} \cdot F_{n}^{c_j}
\end{align*}
and solutions must fulfill a similar distribution constraint:
\begin{align*}
\forall c \in \Cps : D(c) \cdot X \cdot Y &= \sum_{i = 0}^{X-1} \sum_{j = 0}^{Y-1} F_{(i, j)}^c
\end{align*}
\ No newline at end of file
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\contentsline {theorem}{{Theorem}{4.{1}}{}}{7}{theorem.4.2.1}%
\contentsline {proof}{{Proof}{1}{}}{7}{proof.1}%
\contentsline {theorem}{{Theorem}{4.{2}}{}}{9}{theorem.4.3.2}%
\contentsline {proof}{{Proof}{2}{}}{9}{proof.2}%
\contentsline {theorem}{{Theorem}{7.{1}}{}}{19}{theorem.7.2.1}%
\contentsline {proof}{{Proof}{2}{}}{19}{proof.2}%
\contentsline {proof}{{Proof}{3}{}}{19}{proof.3}%
\contentsline {theorem}{{Theorem}{7.{2}}{}}{20}{theorem.7.2.2}%
\contentsline {proof}{{Proof}{3}{}}{21}{proof.3}%
\contentsline {proof}{{Proof}{4}{}}{21}{proof.4}%
\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}%
\contentsline {theorem}{{Theorem}{8.{2}}{}}{27}{theorem.8.1.2}%
\contentsline {proof}{{Proof}{6}{}}{27}{proof.6}%
......@@ -4,11 +4,14 @@
\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 {section}{\numberline {3.1}Integer Version}{5}{section.3.1}%
\contentsline {section}{\numberline {3.2}Fractional Version}{6}{section.3.2}%
\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 {section}{\numberline {4.3}Fractional Pixel Farming is NP Complete}{9}{section.4.3}%
\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}%
......
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