Commit f7d1a134 authored by Michael Keller's avatar Michael Keller
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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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