### Rekrutierung Work

parent bda93dbd
 ... ... @@ -145,35 +145,57 @@ 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 We consider the slightly modified version of the standard 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 a pixel may now contain multiple crops, as explained in the problem definition section. 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. We define the decision version of fractional pixel farming analogously to the standard question: Does a field $F'$ for a relation function $R$ exist such that the score $S(F', R)$ is at least $s$: \begin{align*} s \leq S(F', R) \end{align*} \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. The problem is obviously in NP, as we can non-deterministically choose a field $F'$ for a given score function $R$ and calculate the score in polynomial time in relation to the size of the field and number of crops. In a next step we verify that the solution meets the requirement for crop distribution given to us. This can also be done in polynomial time relational to field size and number of crop types. Finally we test if the score $S(F', R)$ is at least $s$. 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 $G$: regular pixel farming is NP complete. For this purpose we define the constructed pixel farming problem for a given graph $G = (V, E)$ as follows: \begin{align*} X &= \abs{V} && Y = 1\\ \Cps &= V && R(a, b) = \begin{cases} 1 & \{a, b\} \in E\\ 0 & otherwise. \end{cases}\\ s &= 2 \cdot (\abs{V} - 1) && D(c) = \frac{1}{\abs{V}} \quad \forall c \in \Cps\\ F' &\in R_+^{C \times X \times Y} \end{align*} We show that the following implications hold: \begin{enumerate} \item There exists no optimal fractional solution to the constructed pixel farming problem $\implies$ ... ... @@ -184,7 +206,16 @@ this relaxation is also NP complete. \end{enumerate} \end{proof} The first implication is trivial. The first implication is obviously true. We prove this indirectly, that is we show that: There exists a hamiltonian path in $G$'' $\implies$ There exists an optimal fractional solution to the constructed pixel farming problem''. This statement is a direct consequence of the NP completeness proof for the standard pixel farming problem. If a Hamiltonian path exists, then an optimal solution for the standard pixel farming problem instance exists. As this standard solution can also be viewed as an optimal fractional solution ($1$'s and $0$'s are fractions too) we are done. The second implication we prove indirectly. We show instead that the following statement holds: ... ...
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