Commit bda93dbd by Michael Keller

### 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
 ... ... @@ -2,23 +2,26 @@ \BOOKMARK [0][-]{chapter.1}{\376\377\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 2 \BOOKMARK [0][-]{chapter.2}{\376\377\000R\000e\000l\000a\000t\000e\000d\000\040\000W\000o\000r\000k}{}% 3 \BOOKMARK [0][-]{chapter.3}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\000S\000t\000a\000t\000e\000m\000e\000n\000t}{}% 4 \BOOKMARK [0][-]{chapter.4}{\376\377\000C\000o\000m\000p\000u\000t\000a\000t\000i\000o\000n\000a\000l\000\040\000H\000a\000r\000d\000n\000e\000s\000s}{}% 5 \BOOKMARK [1][-]{section.4.1}{\376\377\000T\000h\000e\000\040\000d\000e\000c\000i\000s\000i\000o\000n\000\040\000v\000e\000r\000s\000i\000o\000n\000\040\000o\000f\000\040\000p\000i\000x\000e\000l\000\040\000f\000a\000r\000m\000i\000n\000g}{chapter.4}% 6 \BOOKMARK [1][-]{section.4.2}{\376\377\000P\000i\000x\000e\000l\000\040\000F\000a\000r\000m\000i\000n\000g\000\040\000i\000s\000\040\000N\000P\000\040\000c\000o\000m\000p\000l\000e\000t\000e}{chapter.4}% 7 \BOOKMARK [0][-]{chapter.5}{\376\377\000B\000e\000n\000c\000h\000m\000a\000r\000k\000\040\000P\000r\000o\000b\000l\000e\000m\000s}{}% 8 \BOOKMARK [0][-]{chapter.6}{\376\377\000B\000o\000u\000n\000d\000s}{}% 9 \BOOKMARK [1][-]{section.6.1}{\376\377\000B\000a\000s\000i\000c\000\040\000B\000o\000u\000n\000d}{chapter.6}% 10 \BOOKMARK [1][-]{section.6.2}{\376\377\000G\000\344\000r\000t\000n\000e\000r\000\040\000B\000o\000u\000n\000d}{chapter.6}% 11 \BOOKMARK [1][-]{section.6.3}{\376\377\000L\000P\000\040\000b\000o\000u\000n\000d}{chapter.6}% 12 \BOOKMARK [0][-]{chapter.7}{\376\377\000H\000e\000l\000p\000f\000u\000l\000\040\000S\000t\000a\000t\000e\000m\000e\000n\000t\000s}{}% 13 \BOOKMARK [1][-]{section.7.1}{\376\377\000G\000r\000o\000w\000i\000n\000g\000\040\000s\000o\000l\000u\000t\000i\000o\000n\000s}{chapter.7}% 14 \BOOKMARK [1][-]{section.7.2}{\376\377\000C\000r\000e\000a\000t\000i\000n\000g\000\040\000i\000n\000t\000e\000g\000e\000r\000\040\000s\000o\000l\000u\000t\000i\000o\000n\000s\000\040\000f\000r\000o\000m\000\040\000f\000r\000a\000c\000t\000i\000o\000n\000a\000l\000\040\000s\000o\000l\000u\000t\000i\000o\000n\000s}{chapter.7}% 15 \BOOKMARK [2][-]{subsection.7.2.1}{\376\377\000T\000h\000e\000\040\000s\000t\000a\000n\000d\000a\000r\000d\000\040\000m\000e\000t\000h\000o\000d}{section.7.2}% 16 \BOOKMARK [2][-]{subsection.7.2.2}{\376\377\000T\000h\000e\000\040\000a\000d\000v\000a\000n\000c\000e\000d\000\040\000m\000e\000t\000h\000o\000d}{section.7.2}% 17 \BOOKMARK [0][-]{chapter.8}{\376\377\000T\000h\000e\000\040\000L\000i\000n\000e\000a\000r\000\040\000P\000r\000o\000g\000r\000a\000m\000m\000i\000n\000g\000\040\000M\000e\000t\000h\000o\000d}{}% 18 \BOOKMARK [1][-]{section.8.1}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\000S\000e\000t\000u\000p}{chapter.8}% 19 \BOOKMARK [1][-]{section.8.2}{\376\377\000M\000e\000t\000h\000o\000d}{chapter.8}% 20 \BOOKMARK [0][-]{chapter.9}{\376\377\000G\000r\000a\000d\000i\000e\000n\000t\000\040\000D\000e\000s\000c\000e\000n\000t}{}% 21 \BOOKMARK [0][-]{chapter.10}{\376\377\000C\000o\000n\000c\000l\000u\000s\000i\000o\000n}{}% 22 \BOOKMARK [0][-]{appendix.A}{\376\377\000C\000a\000l\000c\000u\000l\000a\000t\000i\000o\000n\000s\000\040\000A\000p\000p\000e\000n\000d\000i\000x}{}% 23 \BOOKMARK [0][-]{appendix*.4}{\376\377\000B\000i\000b\000l\000i\000o\000g\000r\000a\000p\000h\000y}{}% 24 \BOOKMARK [1][-]{section.3.1}{\376\377\000I\000n\000t\000e\000g\000e\000r\000\040\000V\000e\000r\000s\000i\000o\000n}{chapter.3}% 5 \BOOKMARK [1][-]{section.3.2}{\376\377\000F\000r\000a\000c\000t\000i\000o\000n\000a\000l\000\040\000V\000e\000r\000s\000i\000o\000n}{chapter.3}% 6 \BOOKMARK [0][-]{chapter.4}{\376\377\000C\000o\000m\000p\000u\000t\000a\000t\000i\000o\000n\000a\000l\000\040\000H\000a\000r\000d\000n\000e\000s\000s}{}% 7 \BOOKMARK [1][-]{section.4.1}{\376\377\000T\000h\000e\000\040\000d\000e\000c\000i\000s\000i\000o\000n\000\040\000v\000e\000r\000s\000i\000o\000n\000\040\000o\000f\000\040\000p\000i\000x\000e\000l\000\040\000f\000a\000r\000m\000i\000n\000g}{chapter.4}% 8 \BOOKMARK [1][-]{section.4.2}{\376\377\000P\000i\000x\000e\000l\000\040\000F\000a\000r\000m\000i\000n\000g\000\040\000i\000s\000\040\000N\000P\000\040\000c\000o\000m\000p\000l\000e\000t\000e}{chapter.4}% 9 \BOOKMARK [1][-]{section.4.3}{\376\377\000F\000r\000a\000c\000t\000i\000o\000n\000a\000l\000\040\000P\000i\000x\000e\000l\000\040\000F\000a\000r\000m\000i\000n\000g\000\040\000i\000s\000\040\000N\000P\000\040\000C\000o\000m\000p\000l\000e\000t\000e}{chapter.4}% 10 \BOOKMARK [0][-]{chapter.5}{\376\377\000B\000e\000n\000c\000h\000m\000a\000r\000k\000\040\000P\000r\000o\000b\000l\000e\000m\000s}{}% 11 \BOOKMARK [0][-]{chapter.6}{\376\377\000B\000o\000u\000n\000d\000s}{}% 12 \BOOKMARK [1][-]{section.6.1}{\376\377\000B\000a\000s\000i\000c\000\040\000B\000o\000u\000n\000d}{chapter.6}% 13 \BOOKMARK [1][-]{section.6.2}{\376\377\000G\000\344\000r\000t\000n\000e\000r\000\040\000B\000o\000u\000n\000d}{chapter.6}% 14 \BOOKMARK [1][-]{section.6.3}{\376\377\000L\000P\000\040\000b\000o\000u\000n\000d}{chapter.6}% 15 \BOOKMARK [0][-]{chapter.7}{\376\377\000H\000e\000l\000p\000f\000u\000l\000\040\000S\000t\000a\000t\000e\000m\000e\000n\000t\000s}{}% 16 \BOOKMARK [1][-]{section.7.1}{\376\377\000G\000r\000o\000w\000i\000n\000g\000\040\000s\000o\000l\000u\000t\000i\000o\000n\000s}{chapter.7}% 17 \BOOKMARK [1][-]{section.7.2}{\376\377\000C\000r\000e\000a\000t\000i\000n\000g\000\040\000i\000n\000t\000e\000g\000e\000r\000\040\000s\000o\000l\000u\000t\000i\000o\000n\000s\000\040\000f\000r\000o\000m\000\040\000f\000r\000a\000c\000t\000i\000o\000n\000a\000l\000\040\000s\000o\000l\000u\000t\000i\000o\000n\000s}{chapter.7}% 18 \BOOKMARK [2][-]{subsection.7.2.1}{\376\377\000T\000h\000e\000\040\000s\000t\000a\000n\000d\000a\000r\000d\000\040\000m\000e\000t\000h\000o\000d}{section.7.2}% 19 \BOOKMARK [2][-]{subsection.7.2.2}{\376\377\000T\000h\000e\000\040\000a\000d\000v\000a\000n\000c\000e\000d\000\040\000m\000e\000t\000h\000o\000d}{section.7.2}% 20 \BOOKMARK [0][-]{chapter.8}{\376\377\000T\000h\000e\000\040\000L\000i\000n\000e\000a\000r\000\040\000P\000r\000o\000g\000r\000a\000m\000m\000i\000n\000g\000\040\000M\000e\000t\000h\000o\000d}{}% 21 \BOOKMARK [1][-]{section.8.1}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\000S\000e\000t\000u\000p}{chapter.8}% 22 \BOOKMARK [1][-]{section.8.2}{\376\377\000M\000e\000t\000h\000o\000d}{chapter.8}% 23 \BOOKMARK [0][-]{chapter.9}{\376\377\000G\000r\000a\000d\000i\000e\000n\000t\000\040\000D\000e\000s\000c\000e\000n\000t}{}% 24 \BOOKMARK [0][-]{chapter.10}{\376\377\000C\000o\000n\000c\000l\000u\000s\000i\000o\000n}{}% 25 \BOOKMARK [0][-]{appendix.A}{\376\377\000C\000a\000l\000c\000u\000l\000a\000t\000i\000o\000n\000s\000\040\000A\000p\000p\000e\000n\000d\000i\000x}{}% 26 \BOOKMARK [0][-]{appendix*.4}{\376\377\000B\000i\000b\000l\000i\000o\000g\000r\000a\000p\000h\000y}{}% 27
No preview for this file type
 \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}% ... ...
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!