Commit 6e6da168 by Michael Keller

 ... @@ -155,7 +155,7 @@ convex optimization problem. We show however that ... @@ -155,7 +155,7 @@ convex optimization problem. We show however that this relaxation is also NP complete. this relaxation is also NP complete. We define the decision version of fractional pixel farming We define the decision version of fractional pixel farming analogously to the standard question: Does a field $F'$ analogously to the standard version: Does a field $F'$ for a relation function $R$ exist such that the score for a relation function $R$ exist such that the score $S(F', R)$ is at least $s$: $S(F', R)$ is at least $s$: \begin{align*} \begin{align*} ... @@ -206,22 +206,40 @@ $S(F', R)$ is at least $s$: ... @@ -206,22 +206,40 @@ $S(F', R)$ is at least $s$: \end{enumerate} \end{enumerate} \end{proof} \end{proof} The first implication is obviously true. We prove this indirectly, The first implication is obviously true. We prove this indirectly, that is we show that: There exists a hamiltonian path in $G$'' that is we show that: There exists a hamiltonian path in $G$'' $\implies$ There exists an optimal fractional solution to the $\implies$ There exists an optimal fractional solution to the constructed pixel farming problem''. This statement is a constructed pixel farming problem''. This statement is a direct consequence of the NP completeness proof for the standard direct consequence of the NP completeness proof for the standard pixel farming problem. If a Hamiltonian path exists, then pixel farming problem. If a Hamiltonian path exists, then an optimal solution for the standard pixel farming problem an optimal solution for the standard pixel farming problem instance exists. As this standard solution can also be instance exists. As this standard solution can also be viewed as an optimal fractional solution ($1$'s and $0$'s viewed as an optimal fractional solution ($1$'s and $0$'s are fractions too) we are done. are fractions too) we are done. The second implication we prove indirectly. We show instead The second implication we prove by directly. that the following statement holds: We assume that there exists an optimal solution to the There exists no Hamiltonian path in $G$'' constructed pixel farming problem for a $\implies$ No optimal fractional Graph $G$. An optimal solution to the constructed pixel farming problem exists.'' solution in our case means that every crop likes all of the crops in its neighborhood, otherwise the solution would not obtain the maximal possible score $s$. Further there must also exist a sequence of crops such that: \FloatBarrier \begin{enumerate} \item The sequence only contains each crop exactly once \item Every pixel along the proposed sequence of crops contains some of the proposed crop \end{enumerate} \FloatBarrier If these two facts did not hold, then there must exist a crop within our valid solution that is contained less than once, which would mean our solution is invalid. TODO: MAKE MORE FORMAL As a sequence of all crops where every crop likes its neighbors corresponds to $G$ containing a hamiltonian path, we know our statement must be true. \end{proof} \end{proof} \ No newline at end of file
 \chapter{Related Work} \chapter{Related Work} Here Dr. Arnolds Stuff comes in We are only aware of one other individual \ No newline at end of file who has worked on this problem: Dr. Daniel Arnold from the University of Bern. He was kind enough to share some of the progress he made with us. We summarize the main two methods he developed below. The first method is hill climbing. Here Dr. Arnold proposes a method where the following steps are repeated many times. We start out with an initial field configuration. Then we: \begin{enumerate} \item Propose swapping two pixels \item If the swapping would be beneficial we perform it, otherwise we don't \item Go back to step 1 \end{enumerate} The program terminates when no more beneficial swaps can be performed or it have reached the maximum number of iterations. This method can guarantee that a locally optimal solution was found, which means no single swap would improve the score. However, there are many local optima and thus it is rather unlikely that the global optimum is found with this method. A possible approach for tackling this issue is to rerun the algorithm with different initial field configurations. In practice this is unfortunately not enough to fix the issue on large fields. The second method is simulated annealing. This method is similar to the hill climbing approach with one key difference, we sometimes accept configurations that are worse with a certain probability $p$. The process works as follows. Pick an initial field configuration. Then: \begin{enumerate} \item Propose swapping two pixels \item Calculate the current probability $p$ (see below) \item If the swapping would be beneficial perform it, otherwise swap with probability $p$ \item Go back to step 1 \end{enumerate} where the current swap probability is determined by a Boltzmann distribution: \begin{align*} T_i &= T_0 \cdot K^i\\ \Delta H &= \text{Score change caused by proposed swap}\\ p &= e^{-\Delta H / T} \end{align*} where $T_0$ and $K$ between $0$ and $1$ are constants he determines experimentally. The smaller $K$ is, the faster the method converges. The idea with this method is that the algorithm is allowed so escape local minima in the early stages of the algorithm. If the algorithm accepts a change that makes the score worse it might be able to get "unstuck" from the local optimum it is in and find a solution closer to the global of even the global optimum. The simulated annealing approach appears to work quite well. It by far outperforms the hill climbing approach. We will elaborate further on the specific results of this method in the Benchmark problems section. Finally Dr. Arnold also considers other practical aspects of pixel farming, such as what crops to best replace other with after a cycle of farming is completed. We however won't consider these issues in this paper. \ No newline at end of file