Maximizing photosynthetic outcomes is one of many different objectives of a plant. In this thesis we present/ examine a method to predict an optimal veneation pattern for leafs based on the minimal number of leaf cells that have to be transformed into vein cells to supply the entire leaf with nutrients and water. The model only focusses on the number of cells and disregards other aspects of the vascular system, like the vein hierarchy. To implement this model we used a special variant of the Minimum Dominating Set Problem which we implemented using Integer Linear Programming. We call this variant to model the vascular system the Minimum Connected rooted $k$-hop Dominating Set Problem. Our results show that our implementation is not capable of solving larger instances in a reasonable amount of time. In comparison to an implementation in Answer Set Programming our implementation performs worse using the instances that represent plant leafs. We present a detailled comparison between both versions and tested instances of different structure and size. We analyzed why the Integer Linear Programming implementation performes bad on the leaf graphs. The tests also revealed that on randomly generated graphs the Integer Linear Programming implementation outperformed the Answert Set Programming implemantion.
Maximizing photosynthetic outcomes is one of many different objectives of a plant. In this thesis we present and evaluate a method to predict an optimal venation pattern for leafs based on the minimal number of leaf cells that have to be transformed into vein cells to supply the entire leaf with nutrients and water. The model only focuses on the number of cells and disregards other aspects of the vascular system, like the vein hierarchy. To implement this model we used a special variant of the Minimum Dominating Set Problem which we implemented using Integer Linear Programming. We call this variant to model the vascular system the Minimum Connected rooted $k$-hop Dominating Set Problem. Our results show that our implementation is not capable of solving larger instances in a reasonable amount of time. In comparison to an implementation in Answer Set Programming our implementation performs worse using the instances that represent plant leafs. We present a detailed comparison between both versions and tested instances of different structure and size. We analyzed why the Integer Linear Programming implementation performs bad on the leaf graphs. The tests also revealed that on randomly generated graphs the Integer Linear Programming implementation outperformed the Answer Set Programming implementation.
\caption[Minimum Connected rooted $1$-hop Dominating Set Results on the random graphs]{Minimum Connected rooted $1$-hop Dominating Set Results on the random graphs}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k &\# lazily added constraints & optimal & runtime(s)\\
\hline
GNM\_ 50\_ 122 & 2 & 67 & 11 & 0.03795\\
GNM\_ 50\_ 245 & 2 & 9 & 7 & 0.066219\\
GNM\_ 50\_ 368 & 2 & 0 & 1 & 0.008017\\
GNM\_ 50\_ 490 & 2 & 0 & 1 & 0.002605\\
GNM\_ 50\_ 612 & 2 & 0 & 1 & 0.002223\\
GNM\_ 50\_ 735 & 2 & 0 & 1 & 0.002411\\
GNM\_ 50\_ 858 & 2 & 0 & 1 & 0.002486\\
GNM\_ 50\_ 980 & 2 & 0 & 1 & 0.002173\\
GNM\_ 50\_ 1102 & 2 & 0 & 1 & 0.012025\\
GNM\_ 50\_ 1225 & 2 & 0 & 1 & 0.001756\\
GNM\_ 100\_ 495 & 2 & 6 & 4 & 0.108993\\
GNM\_ 100\_ 990 & 2 & 12 & 2 & 0.060489\\
GNM\_ 100\_ 1485 & 2 & 0 & 1 & 0.022559\\
GNM\_ 100\_ 1980 & 2 & 0 & 1 & 0.004219\\
GNM\_ 100\_ 2475 & 2 & 0 & 1 & 0.004791\\
GNM\_ 100\_ 2970 & 2 & 0 & 1 & 0.044863\\
GNM\_ 100\_ 3465 & 2 & 0 & 1 & 0.004259\\
GNM\_ 100\_ 3960 & 2 & 0 & 1 & 0.004273\\
GNM\_ 100\_ 4455 & 2 & 0 & 1 & 0.003927\\
GNM\_ 100\_ 4950 & 2 & 0 & 1 & 0.003468\\
GNM\_ 250\_ 3112 & 2 & 0 & 2 & 0.270981\\
GNM\_ 250\_ 6225 & 2 & 28 & 1 & 0.101028\\
GNM\_ 250\_ 9338 & 2 & 0 & 1 & 0.17136\\
GNM\_ 250\_ 12450 & 2 & 0 & 1 & 0.031756\\
GNM\_ 250\_ 15562 & 2 & 109 & 1 & 0.257635\\
GNM\_ 250\_ 18675 & 2 & 0 & 1 & 0.035879\\
GNM\_ 250\_ 21788 & 2 & 0 & 1 & 0.030358\\
GNM\_ 250\_ 24900 & 2 & 0 & 1 & 0.024402\\
GNM\_ 250\_ 28012 & 2 & 0 & 1 & 0.018999\\
GNM\_ 250\_ 31125 & 2 & 0 & 1 & 0.016561\\
GNM\_ 500\_ 12475 & 2 & 0 & 2 & 1.123904\\
GNM\_ 500\_ 24950 & 2 & 0 & 1 & 0.663096\\
GNM\_ 500\_ 37425 & 2 & 0 & 1 & 0.228299\\
GNM\_ 500\_ 49900 & 2 & 0 & 1 & 0.272308\\
GNM\_ 500\_ 62375 & 2 & 0 & 1 & 0.29011\\
GNM\_ 500\_ 74850 & 2 & 0 & 1 & 0.249534\\
GNM\_ 500\_ 87325 & 2 & 0 & 1 & 0.250321\\
GNM\_ 500\_ 99800 & 2 & 0 & 1 & 0.170296\\
GNM\_ 500\_ 112275 & 2 & 0 & 1 & 0.148031\\
GNM\_ 500\_ 124750 & 2 & 0 & 1 & 0.119448\\
\end{tabular}
\caption[Minimum Connected rooted $2$-hop Dominating Set Results on the random graphs]{Minimum Connected rooted $2$-hop Dominating Set Results on the random graphs}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k &\# lazily added constraints & optimal & runtime(s)\\
\hline
GNM\_ 50\_ 122 & 3 & 0 & 2 & 0.01651\\
GNM\_ 50\_ 245 & 3 & 0 & 1 & 0.005787\\
GNM\_ 50\_ 368 & 3 & 0 & 1 & 0.007788\\
GNM\_ 50\_ 490 & 3 & 0 & 1 & 0.002089\\
GNM\_ 50\_ 612 & 3 & 0 & 1 & 0.002541\\
GNM\_ 50\_ 735 & 3 & 0 & 1 & 0.00202\\
GNM\_ 50\_ 858 & 3 & 0 & 1 & 0.001855\\
GNM\_ 50\_ 980 & 3 & 0 & 1 & 0.00213\\
GNM\_ 50\_ 1102 & 3 & 0 & 1 & 0.012196\\
GNM\_ 50\_ 1225 & 3 & 0 & 1 & 0.001661\\
GNM\_ 100\_ 495 & 3 & 0 & 1 & 0.026969\\
GNM\_ 100\_ 990 & 3 & 0 & 1 & 0.022669\\
GNM\_ 100\_ 1485 & 3 & 0 & 1 & 0.022822\\
GNM\_ 100\_ 1980 & 3 & 0 & 1 & 0.004204\\
GNM\_ 100\_ 2475 & 3 & 0 & 1 & 0.006448\\
GNM\_ 100\_ 2970 & 3 & 0 & 1 & 0.044946\\
GNM\_ 100\_ 3465 & 3 & 0 & 1 & 0.004356\\
GNM\_ 100\_ 3960 & 3 & 0 & 1 & 0.004163\\
GNM\_ 100\_ 4455 & 3 & 0 & 1 & 0.004094\\
GNM\_ 100\_ 4950 & 3 & 0 & 1 & 0.003533\\
GNM\_ 250\_ 3112 & 3 & 14 & 1 & 0.141794\\
GNM\_ 250\_ 6225 & 3 & 28 & 1 & 0.106819\\
GNM\_ 250\_ 9338 & 3 & 51 & 1 & 0.205765\\
GNM\_ 250\_ 12450 & 3 & 82 & 1 & 0.03714\\
GNM\_ 250\_ 15562 & 3 & 109 & 1 & 0.267159\\
GNM\_ 250\_ 18675 & 3 & 0 & 1 & 0.036207\\
GNM\_ 250\_ 21788 & 3 & 0 & 1 & 0.042911\\
GNM\_ 250\_ 24900 & 3 & 0 & 1 & 0.038669\\
GNM\_ 250\_ 28012 & 3 & 0 & 1 & 0.023179\\
GNM\_ 250\_ 31125 & 3 & 0 & 1 & 0.020695\\
GNM\_ 500\_ 12475 & 3 & 0 & 1 & 0.634489\\
GNM\_ 500\_ 24950 & 3 & 68 & 1 & 0.947696\\
GNM\_ 500\_ 37425 & 3 & 118 & 1 & 0.288719\\
GNM\_ 500\_ 49900 & 3 & 0 & 1 & 0.405276\\
GNM\_ 500\_ 62375 & 3 & 0 & 1 & 0.544754\\
GNM\_ 500\_ 74850 & 3 & 0 & 1 & 0.265611\\
GNM\_ 500\_ 87325 & 3 & 0 & 1 & 0.270045\\
GNM\_ 500\_ 99800 & 3 & 0 & 1 & 0.404701\\
GNM\_ 500\_ 112275 & 3 & 0 & 1 & 0.205316\\
GNM\_ 500\_ 124750 & 3 & 0 & 1 & 0.225787\\
\end{tabular}
\caption[Minimum Connected rooted $3$-hop Dominating Set Results on the random graphs]{Minimum Connected rooted $3$-hop Dominating Set Results on the random graphs}
\end{table}
\subsubsection*{ASP}
\begin{table}[H]
\centering
\begin{tabular}{l ccccccccccccc}
name & k & runtime(s) & optimal\\
\hline
GNM\_ 50\_ 122 & 1 & 0.014 & 11\\
GNM\_ 50\_ 245 & 1 & 0.033 & 7\\
GNM\_ 50\_ 368 & 1 & 0.031 & 5 \\
GNM\_ 50\_ 490 & 1 & 0.050 & 4\\
GNM\_ 50\_ 612 & 1 & 0.055 & 4\\
GNM\_ 50\_ 735 & 1 & 0.044 & 3\\
GNM\_ 50\_ 858 & 1 & 0.050 & 3\\
GNM\_ 50\_ 980 & 1 & 0.059 & 2\\
GNM\_ 50\_ 1102 & 1 & 0.052 & 3\\
GNM\_ 50\_ 1225 & 1 & 0.055 & 1\\
GNM\_ 100\_ 495 & 1 & 32.451 & 14\\
GNM\_ 100\_ 990 & 1 & 278.296 & 8\\
GNM\_ 100\_ 1485 & 1 & 42.545 & 6\\
GNM\_ 100\_ 1980 & 1 & 4.049 & 6\\
GNM\_ 100\_ 2475 & 1 & 0.655 & 4\\
GNM\_ 100\_ 2970 & 1 & 0.226 & 3\\
GNM\_ 100\_ 3465 & 1 & 0.208 & 3\\
GNM\_ 100\_ 3960 & 1 & 0.234 & 2\\
GNM\_ 100\_ 4455 & 1 & 0.253 & 2 \\
GNM\_ 100\_ 4950 & 1 & 0.246 & 1\\
GNM\_ 250\_ 3112 & 1 & 1017.204 & [23;9]\\
GNM\_ 250\_ 6225 & 1 & 1009.124 & [12;6] \\
GNM\_ 250\_ 9338 & 1 & 1009.402 & [8;5]\\
GNM\_ 250\_ 12450 & 1 & 1013.976 & [6;4]\\
GNM\_ 250\_ 15562 & 1 & 1008.099 & [5;4]\\
GNM\_ 250\_ 18675 & 1 & 25.687 & 4\\
GNM\_ 250\_ 21788 & 1 & 1.749 & 3\\
GNM\_ 250\_ 24900 & 1 & 1.830 & 3\\
GNM\_ 250\_ 28012 & 1 & 3.400 & 2\\
GNM\_ 250\_ 31125 & 1 & 1.651 & 1\\
GNM\_ 500\_ 12475 & 1 & 1016.396 & [29;7]\\
GNM\_ 500\_ 24950 & 1 & 1011.967 & [15;4]\\
GNM\_ 500\_ 37425 & 1 & 1010.582 & [10;4]\\
GNM\_ 500\_ 49900 & 1 & 1007.821 & [7;4]\\
GNM\_ 500\_ 62375 & 1 & 1006.141 & [6;4]\\
GNM\_ 500\_ 74850 & 1 & 597.053 & 4\\
GNM\_ 500\_ 87325 & 1 & 621.053 & 4\\
GNM\_ 500\_ 99800 & 1 & 13.348 & 3\\
GNM\_ 500\_ 112275 & 1 & 8.705 & 2\\
GNM\_ 500\_ 124750 & 1 & 8.058 & 1\\
\end{tabular}
\caption[Minimum Connected rooted $1$-hop Dominating Set Results on the random graphs using ASP]{Minimum Connected rooted $1$-hop Dominating Set Results on the random graphs using ASP}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k & optimal & runtime(s)\\
\hline
GNM\_ 50\_ 122 & 2 & 5 & 0.025\\
GNM\_ 50\_ 245 & 2 & 1 & 0.030\\
GNM\_ 50\_ 368 & 2 & 1 & 0.036\\
GNM\_ 50\_ 490 & 2 & 1 & 0.036\\
GNM\_ 50\_ 612 & 2 & 1 & 0.038\\
GNM\_ 50\_ 735 & 2 & 1 & 0.046\\
GNM\_ 50\_ 858 & 2 & 1 & 0.047\\
GNM\_ 50\_ 980 & 2 & 1 & 0.049\\
GNM\_ 50\_ 1102 & 2 & 1 & 0.052\\
GNM\_ 50\_ 1225 & 2 & 1 & 0.048\\
GNM\_ 100\_ 495 & 2 & 4 & 0.084\\
GNM\_ 100\_ 990 & 2 & 2 & 0.098\\
GNM\_ 100\_ 1485 & 2 & 1 & 0.111\\
GNM\_ 100\_ 1980 & 2 & 1 & 0.143\\
GNM\_ 100\_ 2475 & 2 & 1 & 0.151\\
GNM\_ 100\_ 2970 & 2 & 1 & 0.174\\
GNM\_ 100\_ 3465 & 2 & 1 & 0.188\\
GNM\_ 100\_ 3960 & 2 & 1 & 0.206\\
GNM\_ 100\_ 4455 & 2 & 1 & 0.220\\
GNM\_ 100\_ 4950 & 2 & 1 & 0.213\\
GNM\_ 250\_ 3112 & 2 & 2 & 0.521\\
GNM\_ 250\_ 6225 & 2 & 1 & 0.652\\
GNM\_ 250\_ 9338 & 2 & 1 & 0.737\\
GNM\_ 250\_ 12450 & 2 & 1 & 0.867\\
GNM\_ 250\_ 15562 & 2 & 1 & 0.972\\
GNM\_ 250\_ 18675 & 2 & 1 & 1.141\\
GNM\_ 250\_ 21788 & 2 & 1 & 1.221\\
GNM\_ 250\_ 24900 & 2 & 1 & 1.305\\
GNM\_ 250\_ 28012 & 2 & 1 & 1.453\\
GNM\_ 250\_ 31125 & 2 & 1 & 1.519\\
GNM\_ 500\_ 12475 & 2 & 2 & 2.314\\
GNM\_ 500\_ 24950 & 2 & 1 & 2.770\\
GNM\_ 500\_ 37425 & 2 & 1 & 3.236\\
GNM\_ 500\_ 49900 & 2 & 1 & 3.702\\
GNM\_ 500\_ 62375 & 2 & 1 & 4.218\\
GNM\_ 500\_ 74850 & 2 & 1 & 4.799\\
GNM\_ 500\_ 87325 & 2 & 1 & 5.456\\
GNM\_ 500\_ 99800 & 2 & 1 & 6.199\\
GNM\_ 500\_ 112275 & 2 & 1 & 6.268\\
GNM\_ 500\_ 124750 & 2 & 1 & 6.522\\
\end{tabular}
\caption[Minimum Connected rooted $2$-hop Dominating Set Results on the random graphs using ASP]{Minimum Connected rooted $2$-hop Dominating Set Results on the random graphs using ASP}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k & optimal & runtime(s)\\
\hline
GNM\_ 50\_ 122 & 3 & 2 & 0.022\\
GNM\_ 50\_ 245 & 3 & 1 & 0.029\\
GNM\_ 50\_ 368 & 3 & 1 & 0.032\\
GNM\_ 50\_ 490 & 3 & 1 & 0.039\\
GNM\_ 50\_ 612 & 3 & 1 & 0.041\\
GNM\_ 50\_ 735 & 3 & 1 & 0.040\\
GNM\_ 50\_ 858 & 3 & 1 & 0.041\\
GNM\_ 50\_ 980 & 3 & 1 & 0.048\\
GNM\_ 50\_ 1102 & 3 & 1 & 0.051\\
GNM\_ 50\_ 1225 & 3 & 1 & 0.053\\
GNM\_ 100\_ 495 & 3 & 1 & 0.082\\
GNM\_ 100\_ 990 & 3 & 1 & 0.101s\\
GNM\_ 100\_ 1485 & 3 & 1 & 0.119\\
GNM\_ 100\_ 1980 & 3 & 1 & 0.140\\
GNM\_ 100\_ 2475 & 3 & 1 & 0.163\\
GNM\_ 100\_ 2970 & 3 & 1 & 0.172\\
GNM\_ 100\_ 3465 & 3 & 1 & 0.186\\
GNM\_ 100\_ 3960 & 3 & 1 & 0.214\\
GNM\_ 100\_ 4455 & 3 & 1 & 0.227\\
GNM\_ 100\_ 4950 & 3 & 1 & 0.223\\
GNM\_ 250\_ 3112 & 3 & 1 & 0.529\\
GNM\_ 250\_ 6225 & 3 & 1 & 0.657\\
GNM\_ 250\_ 9338 & 3 & 1 & 0.782\\
GNM\_ 250\_ 12450 & 3 & 1 & 0.885\\
GNM\_ 250\_ 15562 & 3 & 1 & 0.967\\
GNM\_ 250\_ 18675 & 3 & 1 & 1.114\\
GNM\_ 250\_ 21788 & 3 & 1 & 1.263\\
GNM\_ 250\_ 24900 & 3 & 1 & 1.323\\
GNM\_ 250\_ 28012 & 3 & 1 & 1.489\\
GNM\_ 250\_ 31125 & 3 & 1 & 1.510\\
GNM\_ 500\_ 12475 & 3 & 1 & 2.297\\
GNM\_ 500\_ 24950 & 3 & 1 & 2.714\\
GNM\_ 500\_ 37425 & 3 & 1 & 3.250\\
GNM\_ 500\_ 49900 & 3 & 1 & 3.719\\
GNM\_ 500\_ 62375 & 3 & 1 & 4.513\\
GNM\_ 500\_ 74850 & 3 & 1 & 4.786\\
GNM\_ 500\_ 87325 & 3 & 1 & 5.305\\
GNM\_ 500\_ 99800 & 3 & 1 & 5.845\\
GNM\_ 500\_ 112275 & 3 & 1 & 6.490\\
GNM\_ 500\_ 124750 & 3 & 1 & 6.802\\
\end{tabular}
\caption[Minimum Connected rooted $3$-hop Dominating Set Results on the random graphs using ASP]{Minimum Connected rooted $3$-hop Dominating Set Results on the random graphs using ASP}
