Commit c0e2ca09 authored by msurl's avatar msurl
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\section{Abstract}\raggedbottom
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.
\pagebreak
Maximizing photosynthetic outcome 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 use 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 reveal that on randomly generated graphs the Integer Linear Programming implementation outperformes the
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\section{Acknowledgement}
I would like to express my gratitude to Prof. Gunnar Klau for giving me the opportunity to write this bachelor thesis and for his support and feedback. I would also like to thank Eline van Mantgem for her invaluable advice and support during the weekly meetings. Finally I would like to thank Sven Schrinner and Philipp Spohr for their extensive feedback that they gave me shortly before the deadline.
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......@@ -2,12 +2,12 @@
\appendix
\section{Appendix}
\subsection*{Full Tables} \label{anhang:zusatz1}
\subsection*{Full Tables} \label{appendix:tables}
\subsubsection*{ILP}
\begin{table}[H]
\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
name & k & \# C & runtime(s) & optimal\\
\hline
GNM\_ 50\_ 122 & 1 & 66 & 0.034878 & 11\\
GNM\_ 50\_ 245 & 1 & 9 & 0.07 & 7\\
......@@ -56,48 +56,48 @@
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k & \# lazily added constraints & optimal & runtime(s)\\
name & k & \# C & runtime(s) & optimal\\
\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\\
GNM\_ 50\_ 122 & 2 & 67 & 0.03795 & 11\\
GNM\_ 50\_ 245 & 2 & 9 & 0.066219 & 7\\
GNM\_ 50\_ 368 & 2 & 0 & 0.008017 & 1\\
GNM\_ 50\_ 490 & 2 & 0 & 0.002605 & 1\\
GNM\_ 50\_ 612 & 2 & 0 & 0.002223 & 1\\
GNM\_ 50\_ 735 & 2 & 0 & 0.002411 & 1\\
GNM\_ 50\_ 858 & 2 & 0 & 0.002486 & 1\\
GNM\_ 50\_ 980 & 2 & 0 & 0.002173 & 1\\
GNM\_ 50\_ 1102 & 2 & 0 & 0.012025 & 1\\
GNM\_ 50\_ 1225 & 2 & 0 & 0.001756 & 1\\
GNM\_ 100\_ 495 & 2 & 6 & 0.108993 & 4\\
GNM\_ 100\_ 990 & 2 & 12 & 0.060489 & 2\\
GNM\_ 100\_ 1485 & 2 & 0 & 0.022559 & 1\\
GNM\_ 100\_ 1980 & 2 & 0 & 0.004219 & 1\\
GNM\_ 100\_ 2475 & 2 & 0 & 0.004791 & 1\\
GNM\_ 100\_ 2970 & 2 & 0 & 0.044863 & 1\\
GNM\_ 100\_ 3465 & 2 & 0 & 0.004259 & 1\\
GNM\_ 100\_ 3960 & 2 & 0 & 0.004273 & 1\\
GNM\_ 100\_ 4455 & 2 & 0 & 0.003927 & 1\\
GNM\_ 100\_ 4950 & 2 & 0 & 0.003468 & 1\\
GNM\_ 250\_ 3112 & 2 & 0 & 0.270981 & 2\\
GNM\_ 250\_ 6225 & 2 & 28 & 0.101028 & 1\\
GNM\_ 250\_ 9338 & 2 & 0 & 0.17136 & 1\\
GNM\_ 250\_ 12450 & 2 & 0 & 0.031756 & 1\\
GNM\_ 250\_ 15562 & 2 & 109 & 0.257635 & 1\\
GNM\_ 250\_ 18675 & 2 & 0 & 0.035879 & 1\\
GNM\_ 250\_ 21788 & 2 & 0 & 0.030358 & 1\\
GNM\_ 250\_ 24900 & 2 & 0 & 0.024402 & 1\\
GNM\_ 250\_ 28012 & 2 & 0 & 0.018999 & 1\\
GNM\_ 250\_ 31125 & 2 & 0 & 0.016561 & 1\\
GNM\_ 500\_ 12475 & 2 & 0 & 1.123904 & 2\\
GNM\_ 500\_ 24950 & 2 & 0 & 0.663096 & 1\\
GNM\_ 500\_ 37425 & 2 & 0 & 0.228299 & 1\\
GNM\_ 500\_ 49900 & 2 & 0 & 0.272308 & 1\\
GNM\_ 500\_ 62375 & 2 & 0 & 0.29011 & 1\\
GNM\_ 500\_ 74850 & 2 & 0 & 0.249534 & 1\\
GNM\_ 500\_ 87325 & 2 & 0 & 0.250321 & 1\\
GNM\_ 500\_ 99800 & 2 & 0 & 0.170296 & 1\\
GNM\_ 500\_ 112275 & 2 & 0 & 0.148031 & 1\\
GNM\_ 500\_ 124750 & 2 & 0 & 0.119448 & 1\\
\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}
......@@ -105,48 +105,48 @@
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k & \# lazily added constraints & optimal & runtime(s)\\
name & k & \# C & runtime(s) & optimal\\
\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\\
GNM\_ 50\_ 122 & 3 & 0 & 0.01651 & 2\\
GNM\_ 50\_ 245 & 3 & 0 & 0.005787 & 1\\
GNM\_ 50\_ 368 & 3 & 0 & 0.007788 & 1\\
GNM\_ 50\_ 490 & 3 & 0 & 0.002089 & 1\\
GNM\_ 50\_ 612 & 3 & 0 & 0.002541 & 1\\
GNM\_ 50\_ 735 & 3 & 0 & 0.00202 & 1\\
GNM\_ 50\_ 858 & 3 & 0 & 0.001855 & 1\\
GNM\_ 50\_ 980 & 3 & 0 & 0.00213 & 1\\
GNM\_ 50\_ 1102 & 3 & 0 & 0.012196 & 1\\
GNM\_ 50\_ 1225 & 3 & 0 & 0.001661 & 1\\
GNM\_ 100\_ 495 & 3 & 0 & 0.026969 & 1\\
GNM\_ 100\_ 990 & 3 & 0 & 0.022669 & 1\\
GNM\_ 100\_ 1485 & 3 & 0 & 0.022822 & 1\\
GNM\_ 100\_ 1980 & 3 & 0 & 0.004204 & 1\\
GNM\_ 100\_ 2475 & 3 & 0 & 0.006448 & 1\\
GNM\_ 100\_ 2970 & 3 & 0 & 0.044946 & 1\\
GNM\_ 100\_ 3465 & 3 & 0 & 0.004356 & 1\\
GNM\_ 100\_ 3960 & 3 & 0 & 0.004163 & 1\\
GNM\_ 100\_ 4455 & 3 & 0 & 0.004094 & 1\\
GNM\_ 100\_ 4950 & 3 & 0 & 0.003533 & 1\\
GNM\_ 250\_ 3112 & 3 & 14 & 0.141794 & 1\\
GNM\_ 250\_ 6225 & 3 & 28 & 0.106819 & 1\\
GNM\_ 250\_ 9338 & 3 & 51 & 0.205765 & 1\\
GNM\_ 250\_ 12450 & 3 & 82 & 0.03714 & 1\\
GNM\_ 250\_ 15562 & 3 & 109 & 0.267159 & 1\\
GNM\_ 250\_ 18675 & 3 & 0 & 0.036207 & 1\\
GNM\_ 250\_ 21788 & 3 & 0 & 0.042911 & 1\\
GNM\_ 250\_ 24900 & 3 & 0 & 0.038669 & 1\\
GNM\_ 250\_ 28012 & 3 & 0 & 0.023179 & 1\\
GNM\_ 250\_ 31125 & 3 & 0 & 0.020695 & 1\\
GNM\_ 500\_ 12475 & 3 & 0 & 0.634489 & 1\\
GNM\_ 500\_ 24950 & 3 & 68 & 0.947696 & 1\\
GNM\_ 500\_ 37425 & 3 & 118 & 0.288719 & 1\\
GNM\_ 500\_ 49900 & 3 & 0 & 0.405276 & 1\\
GNM\_ 500\_ 62375 & 3 & 0 & 0.544754 & 1\\
GNM\_ 500\_ 74850 & 3 & 0 & 0.265611 & 1\\
GNM\_ 500\_ 87325 & 3 & 0 & 0.270045 & 1\\
GNM\_ 500\_ 99800 & 3 & 0 & 0.404701 & 1\\
GNM\_ 500\_ 112275 & 3 & 0 & 0.205316 & 1\\
GNM\_ 500\_ 124750 & 3 & 0 & 0.225787 & 1\\
\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}
......@@ -204,48 +204,48 @@
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k & optimal & runtime(s)\\
name & k & runtime(s) & optimal\\
\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\\
GNM\_ 50\_ 122 & 2 & 0.025 & 5\\
GNM\_ 50\_ 245 & 2 & 0.030 & 1\\
GNM\_ 50\_ 368 & 2 & 0.036 & 1\\
GNM\_ 50\_ 490 & 2 & 0.036 & 1\\
GNM\_ 50\_ 612 & 2 & 0.038 & 1\\
GNM\_ 50\_ 735 & 2 & 0.046 & 1\\
GNM\_ 50\_ 858 & 2 & 0.047 & 1\\
GNM\_ 50\_ 980 & 2 & 0.049 & 1\\
GNM\_ 50\_ 1102 & 2 & 0.052 & 1\\
GNM\_ 50\_ 1225 & 2 & 0.048 & 1\\
GNM\_ 100\_ 495 & 2 & 0.084 & 4\\
GNM\_ 100\_ 990 & 2 & 0.098 & 2\\
GNM\_ 100\_ 1485 & 2 & 0.111 & 1\\
GNM\_ 100\_ 1980 & 2 & 0.143 & 1\\
GNM\_ 100\_ 2475 & 2 & 0.151 & 1\\
GNM\_ 100\_ 2970 & 2 & 0.174 & 1\\
GNM\_ 100\_ 3465 & 2 & 0.188 & 1\\
GNM\_ 100\_ 3960 & 2 & 0.206 & 1\\
GNM\_ 100\_ 4455 & 2 & 0.220 & 1\\
GNM\_ 100\_ 4950 & 2 & 0.213 & 1\\
GNM\_ 250\_ 3112 & 2 & 0.521 & 2\\
GNM\_ 250\_ 6225 & 2 & 0.652 & 1\\
GNM\_ 250\_ 9338 & 2 & 0.737 & 1\\
GNM\_ 250\_ 12450 & 2 & 0.867 & 1\\
GNM\_ 250\_ 15562 & 2 & 0.972 & 1\\
GNM\_ 250\_ 18675 & 2 & 1.141 & 1\\
GNM\_ 250\_ 21788 & 2 & 1.221 & 1\\
GNM\_ 250\_ 24900 & 2 & 1.305 & 1\\
GNM\_ 250\_ 28012 & 2 & 1.453 & 1\\
GNM\_ 250\_ 31125 & 2 & 1.519 & 1\\
GNM\_ 500\_ 12475 & 2 & 2.314 & 2\\
GNM\_ 500\_ 24950 & 2 & 2.770 & 1\\
GNM\_ 500\_ 37425 & 2 & 3.236 & 1\\
GNM\_ 500\_ 49900 & 2 & 3.702 & 1\\
GNM\_ 500\_ 62375 & 2 & 4.218 & 1\\
GNM\_ 500\_ 74850 & 2 & 4.799 & 1\\
GNM\_ 500\_ 87325 & 2 & 5.456 & 1\\
GNM\_ 500\_ 99800 & 2 & 6.199 & 1\\
GNM\_ 500\_ 112275 & 2 & 6.268 & 1\\
GNM\_ 500\_ 124750 & 2 & 6.522 & 1\\
\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}
......@@ -253,48 +253,48 @@
\begin{table}[H]
\centering
\begin{tabular}{l cccccccccccc}
name & k & optimal & runtime(s)\\
name & k & runtime(s) & optimal\\
\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\\
GNM\_ 50\_ 122 & 3 & 0.022 & 2\\
GNM\_ 50\_ 245 & 3 & 0.029 & 1\\
GNM\_ 50\_ 368 & 3 & 0.032 & 1\\
GNM\_ 50\_ 490 & 3 & 0.039 & 1\\
GNM\_ 50\_ 612 & 3 & 0.041 & 1\\
GNM\_ 50\_ 735 & 3 & 0.040 & 1\\
GNM\_ 50\_ 858 & 3 & 0.041 & 1\\
GNM\_ 50\_ 980 & 3 & 0.048 & 1\\
GNM\_ 50\_ 1102 & 3 & 0.051 & 1\\
GNM\_ 50\_ 1225 & 3 & 0.053 & 1\\
GNM\_ 100\_ 495 & 3 & 0.082 & 1\\
GNM\_ 100\_ 990 & 3 & 0.101s & 1\\
GNM\_ 100\_ 1485 & 3 & 0.119 & 1\\
GNM\_ 100\_ 1980 & 3 & 0.140 & 1\\
GNM\_ 100\_ 2475 & 3 & 0.163 & 1\\
GNM\_ 100\_ 2970 & 3 & 0.172 & 1\\
GNM\_ 100\_ 3465 & 3 & 0.186 & 1\\
GNM\_ 100\_ 3960 & 3 & 0.214 & 1\\
GNM\_ 100\_ 4455 & 3 & 0.227 & 1\\
GNM\_ 100\_ 4950 & 3 & 0.223 & 1\\
GNM\_ 250\_ 3112 & 3 & 0.529 & 1\\
GNM\_ 250\_ 6225 & 3 & 0.657 & 1\\
GNM\_ 250\_ 9338 & 3 & 0.782 & 1\\
GNM\_ 250\_ 12450 & 3 & 0.885 & 1\\
GNM\_ 250\_ 15562 & 3 & 0.967 & 1\\
GNM\_ 250\_ 18675 & 3 & 1.114 & 1\\
GNM\_ 250\_ 21788 & 3 & 1.263 & 1\\
GNM\_ 250\_ 24900 & 3 & 1.323 & 1\\
GNM\_ 250\_ 28012 & 3 & 1.489 & 1\\
GNM\_ 250\_ 31125 & 3 & 1.510 & 1\\
GNM\_ 500\_ 12475 & 3 & 2.297 & 1\\
GNM\_ 500\_ 24950 & 3 & 2.714 & 1\\
GNM\_ 500\_ 37425 & 3 & 3.250 & 1\\
GNM\_ 500\_ 49900 & 3 & 3.719 & 1\\
GNM\_ 500\_ 62375 & 3 & 4.513 & 1\\
GNM\_ 500\_ 74850 & 3 & 4.786 & 1\\
GNM\_ 500\_ 87325 & 3 & 5.305 & 1\\
GNM\_ 500\_ 99800 & 3 & 5.845 & 1\\
GNM\_ 500\_ 112275 & 3 & 6.490 & 1\\
GNM\_ 500\_ 124750 & 3 & 6.802 & 1\\
\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}
\end{table}
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\input{results}
\input{discussion}
\input{conclusion}
\input{acknowledgement}
\input{anhang}
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\section{Conclusion}\raggedbottom
Given the fact that we adopted the model from \citep{myky} and only implemented it in another framework the models shortcomings are still present. It disregards different aspects that play a role in the venation pattern for real plants.
\section{Conclusion} \label{section:conclusion}\raggedbottom
Given the fact that we adopted the model from \citep{myky} and only implemented it in another framework, the models shortcomings are still present. It disregards different aspects that play a role in the venation pattern for real plants.
Additionally our implementation, in its current version, is not capable of generating optimal solutions in a reasonable amount of time for the leaf representing graphs. The ASP implementation performs better on these graphs and therefore is the better choice for to implement the model. Even after different approaches to reduce the runtime were evaluated the ASP implementation performed better. Nevertheless there are still approaches that can be evaluated.
Additionally our implementation, in its current version, is not capable of generating optimal solutions in a reasonable amount of time for the leaf representing graphs. The ASP implementation performs better on these graphs and therefore is the better choice to implement the model. Even after different approaches to reduce the runtime have been evaluated the ASP implementation performs better. Nevertheless there are still approaches to improve the ILP version that can be evaluated.
The next step for the ILP implementation should either be to adapt the edge based ILP formulation ESA and aspects of its implementation from the current SCIP-Jack software, or to improve the formulation of this thesis. It propably can be improved by inventing a symmetry breaker that reduces the number of symmetrical unconnected integer solutions which are determined in the solving process. Additionally it should be evaluated which type of constraints can be further preadded that would otherwise be added anyway in the process. Another important point is to find heuristics that allow to determine sufficient lower bounds faster.
Though it is also reasonable to implement the suggestions from \citet{myky} to further improve the ASP implementation as it outperformed the ILP implementation.
Though it is also reasonable to implement the suggestions from \citet{myky} to further improve the ASP implementation as it outperformes the ILP implementation.
\pagebreak
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\section{Implementation} \raggedbottom
Now, we specify the implementation of the ILP-formulations from the Methods section. We implemented the ILP-formulations and Algorithms \ref{alg:addConst} and \ref{alg:minSep} using Python version 3.7.5. As branch and cut framework and MIP-solver we use Gurobi version 9.0.2. Gurobi offers a Python interface called \textit{gurobipy} which can be called from inside python scripts. This interface offers access to functions included in Gurobi.
\section{Implementation} \label{section:implementation} \raggedbottom
Now, we specify the implementation of the ILP formulations from the Methods section. We implemented the ILP-formulations and Algorithms \ref{alg:addConst} and \ref{alg:minSep} using Python version 3.7.5. As branch and cut framework and mixed integer programming solver we use Gurobi version 9.0.2. Gurobi offers a Python interface called \textit{gurobipy} which can be called from inside python scripts. This interface offers access to functions included in Gurobi.
Our implementation is embedded in a conda package. The package is called \textit{k\_ hop\_ dominating\_ set\_ gurobi}. The source of the package can be found on \url{https://gitlab.cs.uni-duesseldorf.de/albi/albi-students/bachelor-mario-surlemont/}.
The package itself can be build via
\begin{lstlisting}[language=bash, frame=none, basicstyle=\small]
conda build .
\end{lstlisting}
after heading into the directory.
To build the package \textit{conda-build} needs to be installed.
To build the package the tool \textit{conda-build} needs to be installed.
Afterwards the package can be installed via
\begin{lstlisting}[language=bash, frame=none]
......@@ -15,7 +15,7 @@ conda install --use-local k_hop_dominating_set_gurobi
It holds the dependencies \textit{networkX}, \textit{matplotlib.pyplot} and \textit{gurobipy}.
The vertex separator constraints as well as the MTZ constraints can be chosen. The choice can be specified via the optional argument \textit{-mtz}, for the use of MTZ-constraints. By default the vertex separators are chosen. If required the additional constraints that have been presented in the method section can also be added to the model via the optional argument \textit{-imn| -rpl| -gaus| -pre} with rpl as abbreviation for the naive constraint to reduce the path length and gaus as abbreviation for the constraint involing the gaussian sum formula. The argument \textit{-pre} adds separators to the model before the solution process is started. When the intermediate node constraint is added via \textit{-imn} the generated solutions might not be optimal anymore.
The vertex separator constraints as well as the MTZ constraints can be chosen. The choice can be specified via the optional argument \textit{-mtz}, for the use of MTZ constraints. By default the vertex separators are chosen. If required the \hyperref[subsection:additional]{additional constraints} that have been presented in the method section can also be added to the model via the optional arguments \textit{-imn| -rpl| -gaus| -pre} with rpl as abbreviation for the naive constraint to reduce the path length and gaus as abbreviation for the constraint involving the gaussian sum formula. The argument \textit{-pre} adds separators to the model before the solution process is started. When the intermediate node constraint is added via \textit{-imn} the generated solutions might not be optimal anymore.
As input networkx graphs stored as ``.graphml'' or ``.gml'' can be used. Also ``.lp'' files from \citep{myky} can be used. A full programm call is
\begin{lstlisting}[language=bash, frame=none, basicstyle=\small]
......@@ -23,7 +23,7 @@ k_hop_dominating_set_gurobi -g graph.graphml -k k [OPTIONS]
\end{lstlisting}
with [OPTIONS] = \{-mtz, -inm, -rpl, -gaus, -pre\}.\\
If the vertex separators are chosen to induce connectivity a lazy approach is used. Gurobi offers a callback function which is called during the solution procedure when different events occur. The function offers a code that communicates the type of the occured event. When the callback code \textit{MIPSOLVE} is communicated a mixed ILP-solution was generated. That is a solution where those variables that must be integers are integers while those variables which do not need to be intergers can be arbitrarily chosen (with respect to the inequalities).
As we only have integer variables in our model the \textit{MIPSOLVE} code tells us that an integer solution $D^*$ was generated. In this case we check whether the graph is connected. We use a function that is included in networkx to check if the graph $G[D^*]$ is connected. If not, algorithm \ref{alg:addConst} is used to add the corresponding constraints.
After a valid solution was found the inputgraph it is plottet via matplotlib.plt. The members of the dominating set are displayed in red while all the other vertices are displayed green.
If the vertex separators are chosen to induce connectivity a lazy approach is used. Gurobi offers a callback function which is called during the solution procedure when different events occur. The function offers a code that communicates the type of the occured event. When the callback code \textit{MIPSOLVE} is communicated a mixed ILP solution has been generated. That is a solution where those variables that must be integers are integers while those variables which do not need to be integers can be arbitrarily chosen (with respect to the inequalities).
As we only have integer variables in our model the \textit{MIPSOLVE} code tells us that an integer solution $D^*$ was generated. In this case we check whether the graph is connected. We use a function that is included in networkx to check if the graph $G[D^*]$ is connected. If not, Algorithm \ref{alg:addConst} is used to add the corresponding constraints.
After a valid solution was found it is plottet via matplotlib.plt. The members of the dominating set are displayed in red while all the other vertices are displayed in green.
The console output shows information about the solving process and the solution. Such as the current upper bound and lower bound.
\ No newline at end of file
\section{Introduction}\raggedbottom
Plants try to optimize their architecture to fulfil different objectives. One of it is to maximize the photosynthetic output. Another one is to minimize the cost to build the vascular system \citep{bio_netw}. To maximize the photosynthetic output plants optimize different parameters. As increasing one parameter can reduce another one, many parameters can not be optimized at the same time \citep{bio_netw} \citep{bio_nutrient}.
Plants try to optimize their architecture to fulfill different objectives. One of it is to maximize the photosynthetic output. Another one is to minimize the cost to build the vascular system \citep{bio_netw}. To maximize the photosynthetic output plants optimize different parameters. As increasing one parameter can reduce another one, many parameters can not be optimized at the same time \citep{bio_netw}, \citep{bio_nutrient}.
In this thesis we focus on one particular mechanism how plants can optimize their photosynthetic output.
To generate photosynthetical gains plants need sunlight, carbondioxid and water. (Photosynthese zitat. )
To generate photosynthetical gains plants need sunlight, carbondioxid and water.
Water and nutrients are supplied via the vascular system. Xylem transports water to the leaves where the mesophyl cells produce sugars. These sugars are carried out to the whole plant by phloem, a tissue specialized on transporting sugars.
Xylem and phloem cells are not able to generate sugars, but they are mandatory to supply water to the mesophyl cells and to transport sugars. To be satisfied with the amount of water mesophyl cells have access to, they must not be more than 2-3 cells away from a xylem cell. In this range water can flow from the xylem cells through mesophyl cells that are not next to a xylem cell via diffusion. At the same time sugars can be transported away from the mesophyl cells and supplied to the phloem if there is a phloem cell in the range of 2-3 cells. (Zitat finden. )
Xylem and phloem cells are not able to generate sugars, but they are mandatory to supply water to the mesophyl cells and to transport sugars. To be satisfied with the amount of water mesophyl cells have access to, they must not be more than two to three cells away from a xylem cell. In this range water can flow from the xylem cells through mesophyl cells that are not next to a xylem cell via diffusion. At the same time sugars can be transported away from the mesophyl cells and supplied to the phloem if there is a phloem cell in the range of two to three cells~\cite[p.~469]{watertransport}.
To produce as much sugar as possible the plant can try to(driven by evolutionary processes) maximize the number of mesophyl cells by minimizing the number of vein cells. In this thesis we describe a method to reproduce an optimal venation pattern that minimizes the number of vein cells with respect to the constraint that all mesophyl cells need to be in a fixed range to vein cells. Leaf veins have a hierarchy. In general there is at least one thick major vein branch and several narrow minor branches. This hierarchy is completely disregarded in our problem formulation. Environmental circumstances also influence the venation pattern \citep{bio_veinh}. These influences on the venation are also completely disregarded in our model.
The input instance is given by an undirected graph $G = (V,E)$ that represents a leaf. The set of vertices $V$ represents the leaf cells while the set of edges $E$ represents the connections between the leaf cells in the form of plasmodesmata. To find an optimal pattern we use a special variant of the dominating set problem. For this problem we present an ILP-formulation and an implementation in a branch and cut framenwork.
To produce as much sugar as possible the plant can try to (driven by evolutionary processes) maximize the number of mesophyl cells by minimizing the number of vein cells. In this thesis we describe a method to reproduce an optimal venation pattern that minimizes the number of vein cells with respect to the constraint that all mesophyl cells need to be in a fixed range to vein cells. Leaf veins have a hierarchy. In general there is at least one thick major vein branch and several narrow minor branches. This hierarchy is completely disregarded in our problem formulation. Environmental circumstances also influence the venation pattern \citep{bio_veinh}. These influences on the venation are also completely disregarded in our model.
The input instance is given by an undirected graph $G = (V,E)$ that represents a leaf. The set of vertices $V$ represents the leaf cells while the set of edges $E$ represents the connections between the leaf cells in the form of plasmodesmata. To find an optimal pattern we use a special variant of the dominating set problem. For this problem we present an Integer Linear Programming (ILP) formulation and an implementation in a branch and cut framenwork.
The dominating set problem and several variants are NP-hard \citep{ilp_np}. For our specific case we demand connectivity between the members of the set. This connectivity in ILP-formulations is subject of different prublications as it is not trivial.
\citet{myky} presented in her bachelors thesis an alternative to ILPs. She implemented an algorithm for our problem using Answer Set Programming (ASP). For larger input instances the ASP-version did not create optimal solutions in a reasonable amount of time. \citet{myky} compared for the case where the dominating set does not need to be connected the runtime from an ILP-Version to the runtime from her ASP-version. Her tests revealed that for this particular problem the ILP-version performed significantly better.
The Dominating Set problem and several variants are NP-hard \citep{ilp_np}. For our specific case we demand connectivity between the members of the set. This connectivity in ILP formulations is subject of different prublications as it is not trivial.
\citet{myky} presents in her bachelors thesis an alternative to ILPs. She implemented an algorithm for our problem using Answer Set Programming (ASP). For larger input instances the ASP version did not create optimal solutions in a reasonable amount of time. \citet{myky} compared the runtime from an ILP version to the runtime from her ASP version for the case where the dominating set does not need to be connected. Her tests reveal that for this particular problem the ILP version performes significantly better.
Goal of this thesis is to formulate an ILP and to evaluate wether if this performs better on our input graphs. We compared the ASP-version with an ILP-formulation that was created in this thesis. Contrary to the presumption that the ILP-version could generate solutions faster, on our input instances the ASP-version was significantly faster.
However the ILP-version outperformed the ASP-version on random graphs. The different characteristics and the runtime for the graphs can be taken from the results section. In the discussion section we discuss which characteristics are responsible for the differences in the runtime and what effect initiates them.
In Section 2, the Preleminaries, we will give a short introduction in ILP. Additionally important defintions are stated. After that in the following Section 3 we define the methods to find an optimal venation pattern. Section 4 demonstrates the implementation. At last in Section 4 and Section 5 we present the results and followed by a discussion on the effectiveness and limitations of the ILP-solution and which characteristics graphs hold to perform either better with the ILP-version or with the ASP-version.
The goal of this thesis is to formulate an ILP and to evaluate wether if this performs better on our input graphs. We compare the ASP version with an ILP formulation that was created in this thesis. Contrary to the presumption that the ILP version could generate solutions faster, on our input instances the ASP version is significantly faster.
However the ILP version outperformes the ASP version on random graphs. The different characteristics and the runtime for the graphs can be taken from the \hyperref[section:results]{results section}. In the \hyperref[section:discussion]{discussion section} we review which characteristics are responsible for the differences in the runtime and what effect initiates them.
In \hyperref[section:preliminaries]{Section 2} we will give a short introduction in ILP. Additionally important defintions are stated. After that in \hyperref[section:methods]{Section 3} we define the methods to find an optimal venation pattern. \hyperref[section:implementation]{Section 4} demonstrates the implementation. At last in \hyperref[section:results]{Section 5} and \hyperref[section:discussion]{Section 6} we present the results followed by a discussion on the effectiveness and limitations of the ILP solution and which characteristics graphs hold to perform either better with the ILP version or with the ASP version.
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\section{Preliminaries}\raggedbottom
\section{Preliminaries}\raggedbottom \label{section:preliminaries}
\subsection{Linear Programming}
Linear programming is a technique to minimize linear functions.
The following definition is based on the book \citep{fischetti2019introduction}\\
The following definition is based on the book \fullcite{fischetti2019introduction} \citep{fischetti2019introduction}.\\
A linear program (LP) problem consists of an linear objective function that is minimized with respect to a set of linear inequalities. \\
A linear program (LP) problem consists of a linear objective function that is minimized with respect to a set of linear inequalities. \\
\\
Linear programs can be expressed as
\[\min\{c^Tx : Ax \geq b, x \geq 0\}\]
where $b \in \mathbb{R}^m$ and $c \in \mathbb{R}^n$ are constant vectors. The matrix $A \in \mathbb{R}^{m \times n}$ contains the coefficients of the $m$ inequalities. We minimize the objective function $c^Tx \in \mathbb{R}$. The vector inequality $Ax \geq b$ has to be satisfied for a valid solution.
The vector $x \in \mathbb{R}^n$ describes possible solutions. If $x \in \mathbb{R}^n$ satisfies all inequalities it is called a feasible solution. A solution $x^*$ is optimal if it respects all inequalities and is minimal.
\\
\\
Integer linear programs (ILPs) are linear programs with the additional restriction that all variables have to be integers: $x \in \mathbb{Z}^n$.
The decision variant of an ILP is NP-complete \citep{ilp_np}.
\\
\\
Each line $j$ of $Ax \geq b$ can be expressed as the sum $\sum_{i=1}^{n}{a_{ij}x_i} \geq b_j$. The objective function can be expressed as $\sum_{i=1}^n{c_ix_i}$. In this thesis we use this notation as we perceive it as more readable.
Each row $j$ of $Ax \geq b$ can be expressed as the sum $\sum_{i=1}^{n}{a_{ij}x_i} \geq b_j$. The objective function can be expressed as $\sum_{i=1}^n{c_ix_i}$. In this thesis we use this notation as we perceive it as more readable.
Combinatorial optimization problems can be modeled with ILPs. Every variable $x_i \in \{0,1\}$ denotes a possible decision to include item $i \in \{1,...,n\}$ in the solution.
\subsection{Definitions}
\begin{definition}[Neighborhood]
Given an undirected graph $G = (V,E)$. Let $N(v)$ denote the neighborhood of a vertex $v$. $N(v)$ can formally be described as follows: \[w \in N(v) \Leftrightarrow \exists (v,w) \in E\]
Given an undirected graph $G = (V,E)$. Let $N(v)$ denote the neighborhood of a vertex $v$. $N(v)$ can formally be described as follows: \[w \in N(v) \Leftrightarrow \{v,w\} \in E\]
\end{definition}
\begin{definition}[Dominating Set]
Given an undirected Graph $G = (V,E)$ a dominating set is a subset $D \subset V$ such that each vertex $v \in V$ is either included in the dominating set or adjacent to at least one vertex which is included in the dominating set. For a dominating set $D$ the following statement is valid
Given an undirected Graph $G = (V,E)$ a dominating set is a subset $D \subset V$ such that each vertex $v \in V$ is either included in the dominating set or adjacent to at least one vertex which is included in the dominating set. For a dominating set $D$ the following expression is valid
\[\forall v \in V \setminus D: \exists u \in D, u \in N(v)\]
\end{definition}
......@@ -32,7 +30,7 @@ The neighborhood of a single vertex $N(v)$ is defined above. Let the neighborhoo
Let $k \in \mathbb{N}$.
With help of this definition the k-neighborhood $N_k(v)$ of a single vertex $v \in V$ can recursively be defined as:
\[N_k(v) := N(N_{k-1}(v)) \setminus v\]
whereas $N_1(v) = N(v)$. So $N_k(v)$ is a set of all vertices which can be reached with at most $k$ steps starting from $v$.
whereas $N_1(v) = N(v)$. This means that $N_k(v)$ is a set of all vertices which can be reached with at most $k$ steps starting from $v$.
\end{definition}
\begin{definition}[k-hop Dominating Set]
......@@ -41,15 +39,13 @@ A $k$-hop dominating set is a subset $D \subset V$ such that for each vertex $v
This means that each vertex is either part of $D$ or in $N_k(w)$ for any $w \in D$.
\end{definition}
\begin{definition}[connected k-hop Dominating Set]
A $k$-hop dominating set $D$ is a connected $k$-hop Dominating Set if the induced subgraph $G[D]$ is connected.
\begin{definition}[Connected k-hop Dominating Set]
A $k$-hop dominating set $D$ is a connected $k$-hop dominating set if the induced subgraph $G[D]$ is connected.
\end{definition}
\begin{definition}[rooted connected k-hop Dominating Set]
\begin{definition}[Rooted Connected k-hop Dominating Set]
Let $v \in V$ be the \emph{root}.
A rooted connected $k$-hop dominating set $D$ is as connected $k$-hop dominating set which also includes $v$.
\end{definition}
(Add a definition for what "connected" means)
\pagebreak
......@@ -52,11 +52,7 @@ doi = {10.1007/s10107-017-1117-8}
}
@book{fischetti2019introduction,
title={Introduction to Mathematical Optimization},
author={Fischetti, M.},
isbn={9781692792022},
url={https://books.google.de/books?id=0sbhyQEACAAJ},
year={2019},
publisher={Independently Published}