From 92771bbac9e175ffa98cb4d21c3eb6362fd9518e Mon Sep 17 00:00:00 2001
From: msurl <masur101@hhu.de>
Date: Fri, 14 Aug 2020 04:53:50 +0200
Subject: [PATCH] update
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Latex/anhang.tex | 310 +++++++-
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Latex/bilder/TestedGraphs.ipe | 16 +-
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.../bilder/vertex_separator_illustration.eps | 203 ++---
.../bilder/vertex_separator_illustration.ipe | 80 +-
Latex/conclusion.tex | 2 +-
Latex/discussion.tex | 17 +-
Latex/implementation.tex | 11 +-
Latex/methods.tex | 36 +-
Latex/preliminaries.tex | 8 +-
Latex/references.bib | 87 +-
Latex/results.tex | 349 ++------
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diff --git a/Latex/abstract.tex b/Latex/abstract.tex
index 880358b..57cf6e6 100644
--- a/Latex/abstract.tex
+++ b/Latex/abstract.tex
@@ -1,3 +1,3 @@
\section{Abstract}\raggedbottom
-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.
\pagebreak
diff --git a/Latex/anhang.tex b/Latex/anhang.tex
index bc840f8..fb904ce 100644
--- a/Latex/anhang.tex
+++ b/Latex/anhang.tex
@@ -1,9 +1,303 @@
-\newpage
-\appendix
-\section{Anhang}
-
-\subsection*{Zusatzteil 1} \label{anhang:zusatz1}
-
-Dies ist ein Anhang.
-
+\newpage
+\appendix
+\section{Appendix}
+
+\subsection*{Full Tables} \label{anhang:zusatz1}
+\subsubsection*{ILP}
+\begin{table}[H]
+\centering
+ \begin{tabular}{l ccccccccccccc}
+ name & k & \# lazily added constraints & runtime(s) & optimal\\
+ \hline
+ GNM\_ 50\_ 122 & 1 & 66 & 0.034878 & 11\\
+ GNM\_ 50\_ 245 & 1 & 9 & 0.07 & 7\\
+ GNM\_ 50\_ 368 & 1 & 0 & 0.013882 & 5 \\
+ GNM\_ 50\_ 490 & 1 & 4 & 0.016478 & 4\\
+ GNM\_ 50\_ 612 & 1 & 0 & 0.017783 & 4\\
+ GNM\_ 50\_ 735 & 1 & 3 & 0.018471 & 3\\
+ GNM\_ 50\_ 858 & 1 & 3 & 0.038161 & 3\\
+ GNM\_ 50\_ 980 & 1 & 3 & 0.023549 & 3\\
+ GNM\_ 50\_ 1102 & 1 & 3 & 0.019566 & 3\\
+ GNM\_ 50\_ 1225 & 1 & 0 & 0.002396 & 1\\
+ GNM\_ 100\_ 495 & 1 & 113 & 0.376731 & 14\\
+ GNM\_ 100\_ 990 & 1 & 17 & 0.488522 & 8\\
+ GNM\_ 100\_ 1485 & 1 & 7 & 0.396982 & 6\\
+ GNM\_ 100\_ 1980 & 1 & 0 & 0.315584 & 5\\
+ GNM\_ 100\_ 2475 & 1 & 0 & 0.045136 & 4\\
+ GNM\_ 100\_ 2970 & 1 & 0 & 0.013737 & 3\\
+ GNM\_ 100\_ 3465 & 1 & 0 & 0.010702 & 3\\
+ GNM\_ 100\_ 3960 & 1 & 0 & 0.007955 & 2\\
+ GNM\_ 100\_ 4455 & 1 & 0 & 0.00505 & 2\\
+ GNM\_ 100\_ 4950 & 1 & 0 & 0.00535 & 1\\
+ GNM\_ 250\_ 3112 & 1 & 0 & 1017.303471 & [17;15]\\
+ GNM\_ 250\_ 6225 & 1 & 0 & 900.64 & 10 \\
+ GNM\_ 250\_ 9338 & 1 & 0 & 29.67 & 7\\
+ GNM\_ 250\_ 12450 & 1 & 0 & 46.78 & 6\\
+ GNM\_ 250\_ 15562 & 1 & 0 & 12.29 & 5\\
+ GNM\_ 250\_ 18675 & 1 & 0 & 0.97 & 4\\
+ GNM\_ 250\_ 21788 & 1 & 3 & 0.415836 & 3\\
+ GNM\_ 250\_ 24900 & 1 & 0 & 0.040482 & 3\\
+ GNM\_ 250\_ 28012 & 1 & 0 & 0.024473 & 2\\
+ GNM\_ 250\_ 31125 & 1 & 0 & 0.017227 & 1\\
+ GNM\_ 500\_ 12475 & 1 & 42 & 1004.920676 & [21;13]\\
+ GNM\_ 500\_ 24950 & 1 & 0 & 1051.277153 & [12;8]\\
+ GNM\_ 500\_ 37425 & 1 & 0 & 9.89 & 4\\
+ GNM\_ 500\_ 49900 & 1 & 0 & 1017.23594 & [6;5]\\
+ GNM\_ 500\_ 62375 & 1 & 0 & 178.495614 & 5\\
+ GNM\_ 500\_ 74850 & 1 & 0 & 9.753998 & 4\\
+ GNM\_ 500\_ 87325 & 1 & 0 & 21.368156 & 4\\
+ GNM\_ 500\_ 99800 & 1 & 0 & 0.286309 & 3\\
+ GNM\_ 500\_ 112275 & 1 & 0 & 0.189313 & 2\\
+ GNM\_ 500\_ 124750 & 1 & 0 & 0.11 & 1\\
+ \end{tabular}
+ \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}
+\end{table}
+
+
\clearpage
\ No newline at end of file
diff --git a/Latex/ba.pdf b/Latex/ba.pdf
new file mode 100644
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diff --git a/Latex/ba.tex b/Latex/ba.tex
index 8e283d5..54cceaa 100644
--- a/Latex/ba.tex
+++ b/Latex/ba.tex
@@ -87,6 +87,7 @@
\input{results}
\input{discussion}
\input{conclusion}
+\input{anhang}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% ENDE TEXTTEIL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/Latex/bilder/TestedGraphs.ipe b/Latex/bilder/TestedGraphs.ipe
index 09c2f94..a4d5f74 100644
--- a/Latex/bilder/TestedGraphs.ipe
+++ b/Latex/bilder/TestedGraphs.ipe
@@ -1,7 +1,7 @@
<?xml version="1.0"?>
<!DOCTYPE ipe SYSTEM "ipe.dtd">
<ipe version="70218" creator="Ipe 7.2.20">
-<info created="D:20200128174124" modified="D:20200804004222"/>
+<info created="D:20200128174124" modified="D:20200812225222"/>
<ipestyle name="basic">
<symbol name="arrow/arc(spx)">
<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
@@ -714,9 +714,9 @@ h
432 672 m
272 512 l
</path>
-<path matrix="1 0 0 1 128 -320" stroke="black">
-400 672 m
-256 528 l
+<path stroke="black">
+528 352 m
+384 208 l
</path>
<path matrix="1 0 0 1 128 -320" stroke="black">
400 704 m
@@ -1679,6 +1679,12 @@ h
<group matrix="1 0 0 1 -176 -320">
<use matrix="1 0 0 1 -152 -24" name="mark/disk(sx)" pos="448 672" size="normal" stroke="green"/>
</group>
-<text matrix="1 0 0 1 112 -432" transformations="translations" pos="0 560" stroke="seagreen" type="label" valign="baseline">Asymmetric</text>
+<text matrix="1 0 0 1 112 -432" transformations="translations" pos="0 560" stroke="seagreen" type="label" width="52.664" height="6.815" depth="1.93" valign="baseline">Asymmetric</text>
+<path stroke="black">
+384 224 m
+384 208 l
+384 208 l
+384 208 l
+</path>
</page>
</ipe>
diff --git a/Latex/bilder/find_minimal_separator_illustration-eps-converted-to.pdf b/Latex/bilder/find_minimal_separator_illustration-eps-converted-to.pdf
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Q Q
showpage
%%Trailer
diff --git a/Latex/bilder/find_minimal_separator_illustration.ipe b/Latex/bilder/find_minimal_separator_illustration.ipe
index 30db6e9..1d801e0 100644
--- a/Latex/bilder/find_minimal_separator_illustration.ipe
+++ b/Latex/bilder/find_minimal_separator_illustration.ipe
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</ipe>
diff --git a/Latex/conclusion.tex b/Latex/conclusion.tex
index 3ffa2b8..7cc7f17 100644
--- a/Latex/conclusion.tex
+++ b/Latex/conclusion.tex
@@ -3,7 +3,7 @@ Given the fact that we adopted the model from \citep{myky} and only implemented
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.
-The next step for the ILP implementation should be to invent a symmetry breaker that reduces the number of symmetrical unconnected integer solutions that are determined in the iteration process. Additionaly 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 suffiecient lower bounds faster.
+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.
\pagebreak
diff --git a/Latex/discussion.tex b/Latex/discussion.tex
index 448b768..545c736 100644
--- a/Latex/discussion.tex
+++ b/Latex/discussion.tex
@@ -1,28 +1,31 @@
\section{Discussion}\raggedbottom
-As already mentioned and as \citet{myky} stated our model has some shortcomings and disregards aspects that influence an optimal venation pattern in real plants. We only focus on minimizing the number of cells that have to be transformed into vein cells, under the condition that the entire leaf can still be supplied with water and nutrients. Doing so the number of photosynthetic active cells and their outcome should be maximized. Our model completely disregards the vein hierarchy and among other things that environmental circumstances also influence the venation pattern \citep{bio_veinh}. The fact that plants try to minimize their total branch length and the transport distance for nutrients \citep{bio_netw} is also disregarded.
+As already mentioned and as \citet{myky} stated our model has some shortcomings and disregards aspects that influence an optimal venation pattern in real plants. We only focus on minimizing the number of cells that have to be transformed into vein cells, under the condition that the entire leaf can still be supplied with water and nutrients. Doing so the number of photo synthetic active cells and their outcome should be maximized. Our model completely disregards the vein hierarchy and among other things that environmental circumstances also influence the venation pattern \citep{bio_veinh}. The fact that plants try to minimize their total branch length and the transport distance for nutrients \citep{bio_netw} is also disregarded.
As our results revealed/ showed the neither the ILP implementation nor the ASP implementation are capable of generating solutions for our leaf graphs in a reasonable amount of time. The ILP implementation is incapable of finding an optimal solution in under 1000 seconds for the instance \textit{middle-leaf}, having only 62 nodes, with parameter $k = 1$. The ASP implementation on the other hand needed only 154 seconds to find an optimal solution. However both version find an appropriate upper bound in less than 1 second. The rest of the solving time is entirely used to close the gap from the lower bound.
-The instance \textit{GNM\_ 500\_ 62375} on the contrary has 500 nodes but the ILP implementation nevertheless finds a solution in 154 seconds, whereas the ASP version could not find an optimal solution after 1000 seconds. As the results show the same difference in runtime on other rather spare and large random graphs the ILP version seems to perform better on random graphs in general. As the results for the random graphs indicated our ILP implementation might be a reasonable approach applied to other problems which can be modelled with the \textit{Minimum Connected (rooted) k-hop Dominating Set} depending on the structure of the input instances.
+The instance \textit{GNM\_ 500\_ 62375} on the contrary has 500 nodes but the ILP implementation nevertheless finds a solution in 154 seconds, whereas the ASP version could not find an optimal solution after 1000 seconds. As the results show the same difference in runtime on other rather spare and large random graphs the ILP version seems to perform better on random graphs in general. As the results for the random graphs indicated our ILP implementation might be a reasonable approach applied to other problems which can be modeled with the \textit{Minimum Connected (rooted) k-hop Dominating Set} depending on the structure of the input instances.
As well as \citet{myky} made the observation for the ASP implementation that an increasing parameter $k$ reduces the runtime significantly our tests showed the same effect using the ILP implementation. For the random graphs and parameter $k = 2$ or $k = 3$ every instance could be solved in less than 1 second. It should also be noted that for most of the instances in this case only a few or even none constraints needed to be added lazily. Optimal solutions consisted in this case for the most instances only of the single root node or contained also a few additional nodes.
These results can not unconditionally applied to other real world problems as their graphs can have specific structures that differ from random graphs.
-Also on our leaf graphs an increasing $k$ implied a better runtime. However in the case of $k = 2$ and $k = 3$ the instances \textit{maple} and \textit{asymmetric} could not be solved under 1000 seconds. We can not simply arbitrarily increase the parameter $k$ in our model as vein cells must be in a range of 2-3 cells from mesophyl cells. \citep{nachschauen_auf_welcher_seite_und_aus_references_übernehmen}.
+Also on our leaf graphs an increasing $k$ implied a better runtime. However in the case of $k = 2$ and $k = 3$ the instances \textit{maple} and \textit{asymmetric} could not be solved under 1000 seconds. We can not simply arbitrarily increase the parameter $k$ in our model as vein cells must be in a range of 2-3 cells from mesophyl cells ~\cite[p.~469]{watertransport}.
The runtime of the grid graphs also went down with increased $k$. For this graphs even with $k = 1$ an optimal solution could be found in under 1000 seconds. Admittedly all instances only had 64 nodes. As for the instance \textit{GRID\_ 8\_ 8} the time to find an optimal solution was 775 seconds it can be assumed that for larger instances the runtime exceeds 1000 seconds.
Using the \textit{intermediate node constraints} reduced the runtime the most. However in the most cases this constraints added unnecessary nodes to a solution which are not included without using this constraint. Nonetheless it could be considered to use this method to create approximative solutions. But for this purpose it would be desirable to formally prove the maximal amount of extra nodes in relation to an optimal solution. However our results show, at least exemplarily, that in most cases even without this additional constraint in rather short time appropriate upper bounds were established.
-For the instance \textit{middle-leaf} for example the ILP implementation as well as the ASP implementation found an upper bound in less than 1 second that does not differ from an optimal solution. Thus an approximation for the upper bound does not seem tobe necessary. In fact a heuristic that generates an appropriate lower bound is much more desirable as closing the gap to the upper bound takes the major amount of time. Even for the rather large instance \textit{maple} an upper bound that does not differ from the optimal solution using the \textit{intermediate node constraint} is found after 29 seconds. At best this constraint could be used to evaluate how good upper bounds from the solving process are. But for this purpose an approximation factor would be necessary. For the \textit{asymmetric} an optimal solution could not be found under 1000 seconds even using this constraint. According to this there is still need for optimisation to create a satisfying implementation even if this constraint is used.
+For the instance \textit{middle-leaf} for example the ILP implementation as well as the ASP implementation found an upper bound in less than 1 second that does not differ from an optimal solution. Thus an approximation for the upper bound does not seem to be necessary. In fact a heuristic that generates an appropriate lower bound is much more desirable as closing the gap to the upper bound takes the major amount of time. Even for the rather large instance \textit{maple} an upper bound that does not differ from the optimal solution using the \textit{intermediate node constraint} is found after 29 seconds. At best this constraint could be used to evaluate how good upper bounds from the solving process are. But for this purpose an approximation factor would be necessary. For the \textit{asymmetric} an optimal solution could not be found under 1000 seconds even using this constraint. According to this there is still need for optimization to create a satisfying implementation even if this constraint is used.
-According to the current information using vertex separators seem to be the best method to induce connectivity on graphtheoretical problems. Alternative approaches from \citep{mtz} or \citep{klau} were not as succesfull for the corresponding problems in comparison to formulations that use vertex separators. Especially for the steiner tree problem \citet{fischetti_steiner_t} could achieve good results compared to other approaches. Also \citet{bomersbach} could achieve good results for the Connected Maximum Coverage Problem. In \citep{forrest} and \citep{fault_tolerant} this method was evaluated as promising.
+According to the current information using vertex separators seem to be the best method to induce connectivity on graph theoretical problems. Alternative approaches from \citep{mtz} or \citep{klau} were not as succesfull for the corresponding problems in comparison to formulations that use vertex separators. Especially for the steiner tree problem \citet{fischetti_steiner_t} could achieve good results compared to other approaches. Also \citet{bomersbach} could achieve good results for the Connected Maximum Coverage Problem. In \citep{forrest} and \citep{fault_tolerant} this method was evaluated as promising.
For our problem and especially for the graphs that represent our leafs this method was not satisfying. The same applies to quadratical grid graphs. We assume the high number of unconnected integer solutions that are generated in the iteration process as beeing crucial. These solutions are most likely in some manner symmetrical such that an appropriate symmetry breaker could reduce the runtime drastically.
In general the ASP implementation performed better on our graphs representing the leafs. \citet{myky} mentioned different aspects in the conclusion of her thesis how the ASP implementation can be improved. As this implementation performed better than the ILP implementation so far it might be more reasonable to improve the ASP implementation rather than the ILP.
-Another aspect that our tests revealed is that espacially on such instance where there is a rather large gap between the size of an optimal unconnected solution and an optimal connected solution the runtime is relatively high. This is probably related to the fact that in such cases many constraints were added lazily, which indicates that there is a high amount of unconnected integer solutions. For the instances where the gap was rather tight the runtime was much better. In the tests from \citep{myky} an ILP implementation for the unconnected Minimum $k$-hop Dominanting Set could create solutions much faster than the ASP implementation. This specific superiority is reflected here such that quickly valid solutions could be generated and it only neede to be verified if the solution is connected and otherwise only a few constraints needed to be added.
+Another aspect that our tests revealed is that especially on such instance where there is a rather large gap between the size of an optimal unconnected solution and an optimal connected solution the runtime is relatively high. This is probably related to the fact that in such cases many constraints were added lazily, which indicates that there is a high amount of unconnected integer solutions. For the instances where the gap was rather tight the runtime was much better. In the tests from \citep{myky} an ILP implementation for the unconnected Minimum $k$-hop Dominating Set could create solutions much faster than the ASP implementation. This specific superiority is reflected here such that quickly valid solutions could be generated and it only needs to be verified if the solution is connected and otherwise only a few constraints needed to be added.
-The density has also shown as a parameter which highly influences the runtime. On sparse graphs both the ILP implementation and the ASP implementation performed rather bad. For the random graphs instances with 250 and 500 nodes coould not be solved under 1000 seconds on rather sparse graphs with parameter $k = 1$. Our leaf graphs are all very sparse such that this effect plays a role as well. With increasing size the densitiy of our graphs even decreases.
+The density has also shown as a parameter which highly influences the runtime. On sparse graphs both the ILP implementation and the ASP implementation performed rather bad. For the random graphs instances with 250 and 500 nodes could not be solved under 1000 seconds on rather sparse graphs with parameter $k = 1$. Our leaf graphs are all very sparse such that this effect plays a role as well. With increasing size the density of our graphs even decreases.
Preadding vertex separator constraints had an measurable influence on the runtime. Unfortunately this effect alone could not improve the runtime in a manner that a satisfying implementation for our model could be created. Despite the fact that many constraints were preadded there were still a lot constraints that were added in the iteration process. It could make sense to identify the types of constraints that are still added in the solution process to prevent unnecessary iterations when they are added beforehand. This might lead to a better runtime.
Another approach to improve the implementation can be to add violated constraints not only after integer solutions are created but already when LP relaxations are calculated. This approach was used in \citep{forrest} and lead to sufficient LP bounds.
Eine weitere Möglichkeit, das Verfahren zu optimieren, wäre es, constraints nicht nur dann hinzuzufügen, wenn eine ganzzahlige Lösung ermittelt wurde, sondern schon dann, wenn eine LP relaxierung ermittelt wird. Dieser Ansatz wurde auch in \citep{forrest} verfolgt. Dabei konnten sehr gute Erfolge hinsichtlich der Lp Bound erzielt werden.
+
+Recently a paper was published that compared different ILP formulations for the MWCSP \citep{esa}. \citet{esa} compare theoretically and empirically an edge based ILP formulation called Extended Steiner Arborescence Formulation (ESA) with the ILP formulation from \citep{fischetti_steiner_t}. In this paper it is proven that the polyhedron of the ESA is a real subset of the node based formulation from \citep{fischetti_steiner_t}. The computational results show that the ESA outperforms the node based one as the runtime was shorter for most instances. Also the with the ESA it was possible to solve previous unsovled instances. It it possible to create an ILP formulation for our model which uses the connectivity inducing constraints of the ESA. The implementation of the ESA is embedded in the upcoming version of \textit{SCIP-Jack}, a \textbf{C} based branch-and-cut framework for the Steiner Tree problem. The ESA also needs exponentially many constraints to induce connectivity. As the efficiency and the runtime of a branch-and-cut approach depends on concrete implementation details and used heuristics, it would be necessary to explore the source code and the documentation. There are also several publications that can be found on the official SCIP webpage \url{https://www.scipopt.org/} that can be helpful.
+It could also be reasonable to combine both approaches, such that a minimum $k$-hop dominating set $D_t$ is found at first and afterwards a minimun weight connected steiner tree $D$ with $D_t$ as set of terminals is found. This method could benefit from the facts that our ILP formulation can find unconnected minimum $k$-hop dominating sets rather quickly and MWCST instances can be solved very quick using ESA. However this might lead to not necessarily optimal solutions.
\pagebreak
diff --git a/Latex/implementation.tex b/Latex/implementation.tex
index 0ccbd7a..988a643 100644
--- a/Latex/implementation.tex
+++ b/Latex/implementation.tex
@@ -5,7 +5,7 @@ The package itself can be build via
\begin{lstlisting}[language=bash, frame=none, basicstyle=\small]
conda build .
\end{lstlisting}
-After heading into the directory.
+after heading into the directory.
To build the package \textit{conda-build} needs to be installed.
Afterwards the package can be installed via
@@ -19,10 +19,11 @@ The vertex separator constraints as well as the MTZ constraints can be chosen. T
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]
-k_hop_dominating_set_gurobi (-mtz) (-inm) (-rpl) (-gaus) (-pre) graph.graphml k
+k_hop_dominating_set_gurobi -g graph.graphml -k k [OPTIONS]
\end{lstlisting}
-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 an 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 if 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 is plottet via matplotlib.plt. The members of the dominating set are displayed red while all the other vertices are displayed green.
+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.
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
diff --git a/Latex/methods.tex b/Latex/methods.tex
index 2fa1575..0878cb4 100644
--- a/Latex/methods.tex
+++ b/Latex/methods.tex
@@ -36,7 +36,7 @@ The algorithm from \citep{fischetti_steiner_t} significantly improved the runtim
\begin{figure}
\centering
\includegraphics[width=10cm]{bilder/vertex_separator_illustration.eps}
- \caption{Illustration of vertex separators. In all three pictures the set of green nodes separates the blue and the red node. In the middle and on the right picture minimal separators are illustrated. If one of the green nodes is turned into a black node, the green set would not separate the blue and the red node anymore. }
+ \caption[Illustration of vertex separators]{Illustration of vertex separators. In all three pictures the set of green nodes separates the blue and the red node. In the middle and on the right picture minimal separators are illustrated. If one of the green nodes is turned into a black node, the green set would not separate the blue and the red node anymore. }
\label{mtz}
\end{figure}
@@ -55,7 +55,7 @@ x_v + x_w \leq \sum_{u \in S_{v,w}}{x_u} + 1, \forall v, w \in V, v \neq w, \for
\end{equation}
for minimum vertex separators that include the root node.
-The number of all minimum vertex seperator constraints is potentially exponential \citep{bomersbach}. Therefore in \citep{bomersbach}, \citep{fischetti_steiner_t} and \citep{forrest} they treated these constraints as lazy constraints, which means in particular that none of those constraints are included in the initial model. Instead iteratively integer solutions are resolved \citep{bomersbach}, \citep{fischetti_steiner_t}. If such a solution is not connected, in \citep{bomersbach} and \citep{fischetti_steiner_t} minimal vertex separators that separate single components are identified via a linear time algorithm, while in \citep{forrest} a classical max-flow min-cut theorem is used to identify violated constraints.\\
+The number of all minimum vertex separator constraints is potentially exponential \citep{bomersbach}. Therefore in \citep{bomersbach}, \citep{fischetti_steiner_t} and \citep{forrest} they treated these constraints as lazy constraints, which means in particular that none of those constraints are included in the initial model. Instead iteratively integer solutions are resolved \citep{bomersbach}, \citep{fischetti_steiner_t}. If such a solution is not connected, in \citep{bomersbach} and \citep{fischetti_steiner_t} minimal vertex separators that separate single components are identified via a linear time algorithm, while in \citep{forrest} a classical max-flow min-cut theorem is used to identify violated constraints.\\
Our algorithm to identify and add violated constraints is analogous the one from \citep{bomersbach} with the exception that we only search for violated constraints that include the root node.
\begin{algorithm}[H] \label{alg:addConst}
@@ -79,7 +79,7 @@ Our algorithm to identify and add violated constraints is analogous the one from
}
\caption{Add violated constraints}
\end{algorithm}
-This algorithm is executed each time an integer solution is resolved (using a branch and cut framework). Let $D^*$ be an integer, not necessarily connected, solution. Let $C$ be the set of all connected components from the graph $G' = G[D^*]$ and let $c_r$ be the component that contains the root node $v_r$. Then the algorithm detects for all single components $c \in C \setminus \{c_r\}$ one minimal vertex separator that separates $c$ and the component $c_r$. The constraints concerning these separators are then added to the model and the branch and cut procedure continues. It is important to mention that there is in general more than one minimal vertex separator which seperates two arbitrary components. The algorithm \ref{alg:minSep} detects exactly one, i.e., the separator, that is closest to the first component. By executing the algorithm \ref{alg:minSep} with every component $c \in C \setminus \{c_r\}$ as first component and $c_r$ as second component and vice versa, we ensure that a minimal vertex separator that is closest to each of the components is added.
+This algorithm is executed each time an integer solution is resolved (using a branch and cut framework). Let $D^*$ be an integer, not necessarily connected, solution. Let $C$ be the set of all connected components from the graph $G' = G[D^*]$ and let $c_r$ be the component that contains the root node $v_r$. Then the algorithm detects for all single components $c \in C \setminus \{c_r\}$ one minimal vertex separator that separates $c$ and the component $c_r$. The constraints concerning these separators are then added to the model and the cutting plane procedure continues. It is important to mention that there is in general more than one minimal vertex separator which separates two arbitrary components. The Algorithm \ref{alg:minSep} detects exactly one, i.e., the separator, that is closest to the first component. By executing the Algorithm \ref{alg:minSep} with every component $c \in C \setminus \{c_r\}$ as first component and $c_r$ as second component and vice versa, we ensure that a minimal vertex separator that is closest to each of the components is added.
\begin{algorithm}[H] \label{alg:minSep}
\SetAlgoLined
@@ -90,7 +90,7 @@ This algorithm is executed each time an integer solution is resolved (using a br
\caption{findMinVertexSeparator($G$, $DS^*$, $v \in c_v$, $w$, $c_v$)}
\end{algorithm}
-The algorithm above detects a minimal vertex separator that seperates the node $w$ and the connected component $c_v$. It is taken from \citep{bomersbach} although \citet{bomersbach} took it initially from \citep{fischetti_steiner_t}. With this method the minimal vertex separator is found that is closest to the component $c_v$. In picture \ref{pic:min_sep} one can see an illustration of the process. Suppose the red marked nodes are an unconnected solution $D^*$. The set of blue marked nodes is the minimal separator that is closest to the connected component on the upper graph while the set of green marked nodes is the minimal separator that is closest to the component containing the root. On the picture in the middle and the right you can see the step \ref{remEdges} of the algorithm \ref{alg:minSep}. As one can see, after removing all edges between the components and its neighborhood the blue marked nodes on the middle picture and the green marked nodes on the right picture are still reachable from the other component. Therefore the algorithm returns this selection of nodes as minimal vertex separator.
+The algorithm above detects a minimal vertex separator that separates the node $w$ and the connected component $c_v$. It is taken from \citep{bomersbach} although \citet{bomersbach} took it initially from \citep{fischetti_steiner_t}. With this method the minimal vertex separator is found that is closest to the component $c_v$. In figure \ref{pic:min_sep} one can see an illustration of the process. Suppose the red marked nodes are an unconnected solution $D^*$. The set of blue marked nodes is the minimal separator that is closest to the connected component on the upper graph while the set of green marked nodes is the minimal separator that is closest to the component containing the root. On the picture in the middle and the right one can see the line \ref{alg:remEdges} of the algorithm \ref{alg:minSep}. As one can see, after removing all edges between the components and its neighborhood the blue marked nodes on the middle picture and the green marked nodes on the right picture are still reachable from the other component. Therefore the algorithm returns this selection of nodes as minimal vertex separator.
\begin{figure}
\centering
@@ -99,22 +99,22 @@ The algorithm above detects a minimal vertex separator that seperates the node $
\label{pic:min_sep}
\end{figure}
-We add an additonal constraint to the model to tighten up the feasible region and to prevent unnecessary iterations.
+We add an additional constraint to the model to tighten up the feasible region and to prevent unnecessary iterations.
\begin{equation} \label{neigh}
x_v \leq \sum_{w \in N(v)} x_w, \forall v \in V \setminus \{v_{root}\}
\end{equation}
This constraint demands that for each vertex which is part of the dominating set at least one of its neighbors is also included. In \citep{bomersbach} and \citep{fischetti_steiner_t} this constraint is also part of the model. As the neighborhood of a single vertex is always a minimal vertex separator that separates this node from any other vertex outside the neighborhood, this constraint is valid. We exclude the root node $v_{root}$ to prevent that for the case of a valid solution that only contains $1$ single vertex another one is added unnecessarily.
\subsubsection{Miller-Tucker-Zemlin Constraints}
-There are also formulations to enforce connectivity that only need a polynomial number of constraints. These constraints are not added lazily but instead all added initially. There exist some approaches that base on the construction of a spanning tree. We have implemented one of these formulations in the scope of this thesis. This approach was used in \citep{mtz} to generate an ILP-formulation for the Minimum Connected Dominating Set problem. In the scope of the publication $4$ different formulations, all based on creating a spanning tree, were compared (experimentally). This particular formulation outperformed all $3$ others on all $6$ inputgraphs. With increasing size the difference in the runtime became larger.
+There are also formulations to enforce connectivity that only need a polynomially number of constraints. These constraints are not added lazily but instead all added initially. There exist some approaches that are based on the construction of a spanning tree. We have implemented one of these formulations in the scope of this thesis. This approach was used in \citep{mtz} to generate an ILP-formulation for the Minimum Connected Dominating Set problem. In the scope of the publication four different formulations, all based on creating a spanning tree, were compared (experimentally). This particular formulation outperformed all three others on all six input graphs. With increasing size the difference in the runtime became larger.
In the scope of this thesis we therefore only compared this one with the vertex separator version.
-The Miller Tucker Zemlin constraints were initially introduced to present an ILP-formulation for the Traveling Salesman Problem with only polynomial many constraints. Let $G =(V,E)$ be our undirected inputgraph. We follow the description from \citep{mtz} by defining $G_d = (V \cup \{n+1, n+2\}, A)$ as directed graph, whereas $A = \{(n+1, n+2)\} \cup \{\bigcup_{i=1}^n{(n+1,i), (n+2,i)} \} \cup E'$ and $E' = \{(j,i), (i,j): {i,j} \in E \}$. Note that $E'$ is the bidirected version of $E$, that means, we add an arc in both directions for every edge in $E$.
+The Miller Tucker Zemlin constraints were initially introduced to present an ILP-formulation for the Traveling Salesman Problem with only polynomial many constraints. Let $G =(V,E)$ be our undirected input graph. We follow the description from \citep{mtz} by defining $G_d = (V \cup \{n+1, n+2\}, A)$ as directed graph, whereas $A = \{(n+1, n+2)\} \cup \{\bigcup_{i=1}^n{(n+1,i), (n+2,i)} \} \cup E'$ and $E' = \{(j,i), (i,j): {i,j} \in E \}$. Note that $E'$ is the bidirected version of $E$, that means, we add an arc in both directions for every edge in $E$.
Let $n = |V|$. We create two additional nodes $n+1$ and $n+2$. Additionally we add an arc from $n+1$ and from $n+2$ to every vertex $v \in V$, and we add an arc from $n+1$ to $n+2$.
The idea behind the constraints is to create a directed spanning tree $T_d = (V \cup {n+1,n+2}, E_d)$ on $G_d$, such that vertex $n+1$ is a root and holds an arc (on $T_d$) to every vertex, which is not part of $D$ and to $n+2$. While $n+2$ holds an arc to a node $v_r$ within $D$. All the other nodes form a tree with root $v_r$.
-Let $y_{ij} \forall (i,j) \in A$ be decision variables, that specify whether the arc $(i,j)$ is part of the spanning tree $T_d$. Let $u_i \in \mathbb{Z}_+, \forall i \in V \cup \{n+1, n+2\}$ be auxilliary variables, that specify in which step the arc is passed starting from $n+1$. Those auxilliary variables eliminate subtours as they also do in the Traveling Salesman Problem.
+Let $y_{ij} \forall (i,j) \in A$ be decision variables, that specify whether the arc $(i,j)$ is part of the spanning tree $T_d$. Let $u_i \in \mathbb{Z}_+, \forall i \in V \cup \{n+1, n+2\}$ be auxiliary variables, that specify in which step the arc is passed starting from $n+1$. Those auxiliary variables eliminate sub tours as they also do in the Traveling Salesman Problem.
In the following we give a full ILP-formulation for to enforce connectivity via MTZ-constraints.
@@ -154,8 +154,8 @@ x_i = 1-y_{n+1,i}, \forall i \in V
\end{figure}
Constraints \eqref{mtz_eq_1} ensure that there is exactly one root for the dominating set. In our case we replace this inequality by the following: $y:{n+2,v_{root}} = 1$ and $y_{n+2, i} = 0, \forall i \in V \setminus \{v_{root}\}$.
-Constraints \eqref{mtz_eq_2} enforce that each node on the spanning tree $T_d$ has exactly one incoming arc. While constraints \eqref{mtz_eq_3} require that all the nodes from $T_d$ are either connected to each other or have an incoming arc from node $n+1$, the node which marks nodes that are not part of $D$. With the exception of the term $(n-1)y_{ji}$ the constraints \eqref{mtz_eq_4} and \eqref{mtz_eq_5} are the original MTZ constraints to eliminate subtours from \citep{mtz_orig}. The mentioned term is an improvement from \citep{mtz_improv}. Constraint \eqref{mtz_eq_6} demands the arc $(n+1,n+2)$ to be included in $T_d$.
-Constraints \eqref{mtz_eq_8} define the value of ranges for the auxilliary variables $u_i$. As these variables specify in which step the arc to node $i$ is passed, only values from $1$ - $n+1$ (the number of incoming arcs) can be assigned to it. Finally the last constraints \eqref{mtz_eq_9} ensure that if there is no incoming arc from node $n+1$ to a node $i$, then $i$ must be included in $D$ and vice versa (I think it is important to mention the backward direction as otherwise the impression could arise that only the MTZ constraints decide which vertices are included).
+Constraints \eqref{mtz_eq_2} enforce that each node on the spanning tree $T_d$ has exactly one incoming arc. While constraints \eqref{mtz_eq_3} require that all the nodes from $T_d$ are either connected to each other or have an incoming arc from node $n+1$, the node which marks nodes that are not part of $D$. With the exception of the term $(n-1)y_{ji}$ the constraints \eqref{mtz_eq_4} and \eqref{mtz_eq_5} are the original MTZ constraints to eliminate sub tours from \citep{mtz_orig}. The mentioned term is an improvement from \citep{mtz_improv}. Constraint \eqref{mtz_eq_6} demands the arc $(n+1,n+2)$ to be included in $T_d$.
+Constraints \eqref{mtz_eq_8} define the value of ranges for the auxiliary variables $u_i$. As these variables specify in which step the arc to node $i$ is passed, only values from $1$ - $n+1$ (the number of incoming arcs) can be assigned to it. Finally the last constraints \eqref{mtz_eq_9} ensure that if there is no incoming arc from node $n+1$ to a node $i$, then $i$ must be included in $D$ and vice versa (I think it is important to mention the backward direction as otherwise the impression could arise that only the MTZ constraints decide which vertices are included).
We combine the above mentioned ILP-formulation for MkCDS with this formulation to enforce connectivity. The solution of this formulation then is a optimal connected solution with $v \in D \Leftrightarrow x_v = 1$. As previously mentioned this formulation only needs polynomial many constraints. More precisely there are $(|V|+2) + (2|E|+2|V|+1) = O(|E|+|V|)$ decision variables and $1 + |V| + 2|E| + 2|E| + (2|V|+1) + 1 + 1 + |V| = O(|E|+|V|)$ constraints.
@@ -169,10 +169,10 @@ x_{v_{root}} \geq 1
\end{equation}
\subsection{Additional methods to tighten up the space of feasible solutions}
-In the scope of this thesis additional contraints were tested, that should tighten up the space of feasable solutions further. As it can potentially cost much time to create unconnected solutions, we want to prevent unnecessary iterations.
+In the scope of this thesis additional constraints were tested, that should tighten up the space of feasible solutions further. As it can potentially cost much time to create unconnected solutions, we want to prevent unnecessary iterations.
\subsubsection{Intermediate node constraint}
-In the paper about the Steiner Tree Problem \citep{fischetti_steiner_t} one inequality to reduce the number of unconnected feasible solutions is proposed. It demands that for each node in the solution, which is not a predefined terminal, to have two neighhbors in the solution. A node that has two neighbors in the solution can be seen as an intermediate node. Let $T$ be the set of all terminals. The inequality can formaly be described as
-\[2 * x_v \leq \sum_{w \in N(v)}{x_v}, \forall v \in T\].
+In the paper about the Steiner Tree Problem \citep{fischetti_steiner_t} one inequality to reduce the number of unconnected feasible solutions is proposed. It demands that for each node in the solution, which is not a predefined terminal, to have two neighbors in the solution. A node that has two neighbors in the solution can be seen as an intermediate node. Let $T$ be the set of all terminals. The inequality can formally be described as
+\[2 * x_v \leq \sum_{w \in N(v)}{x_v}, \forall v \in T\]
Unfortunately this inequality can not be applied to our problem without potentially excluding optimal solutions. By this inequality solutions can be generated, which have additional nodes at the end of branches, that are not necessary for the MkCDS but that are necessary to fulfill this inequality. In our case we would need to require that for each vertex, which is not at the end of a branch, this inequality needs to be satisfied. But we can not decide which node will be at the end of a branch in advance.
\begin{figure}
@@ -182,17 +182,17 @@ Unfortunately this inequality can not be applied to our problem without potentia
\label{pic:inc}
\end{figure}
-In \eqref{pic:inc} there is an illustration that compares one optimal solution without this constraint on the left and one with this constraint on the right. On the right hand side the end of a branch is circled to outline the additional node generated by this constraint.
+In figure \eqref{pic:inc} there is an illustration that compares one optimal solution without this constraint on the left and one with this constraint on the right. On the right hand side the end of a branch is circled to outline the additional node generated by this constraint.
-Even if the generated solutions are not inevitably optimal, the generated solutions are close to an optimal solution (in terms of the number of nodfes). At the same time this constraint reduces the runtime in many instances drastically. That is why it can be considered to generate approximative solutions using this constraint. This constraint can also be used to generate a sufficient upper bound in the branch and cut process. But for the most instances this is not necessary as a sufficient upper bound is found quickly. It needs much more time to find a sufficient lower bound and to close the gap.
+Even if the generated solutions are not inevitably optimal, the generated solutions are close to an optimal solution (in terms of the number of nodes). At the same time this constraint reduces the runtime in many instances drastically. That is why it can be considered to generate approximative solutions using this constraint. This constraint can also be used to generate a sufficient upper bound in the branch and cut process. But for the most instances this is not necessary as a sufficient upper bound is found quickly. It needs much more time to find a sufficient lower bound and to close the gap.
\subsubsection{Reduce path length}
To exclude such solutions which contain single (unconnected) nodes, that are close to the rim we invented constraints to reduce the length of each path between the nodes of a solution and the root node. The length of each path to an arbitrary node is naturally limited by the number of members of the dominating set. In the extreme there is one single branch, which has exactly the length of the number of all members of the dominating set. In the case of more than one branch the upper bound is still valid. On that account we started by following the naive approach to limit the path from the root node to each member of $D$ by the size of $D$. The formal description is
\begin{equation} \label{gaussian}
\sum_{v \in V}{x_v} \geq shortestpath\{v_{root}, v\}, \forall v \in V \setminus \{v_{root}\}
-\end{equation}.
-As this constraint did not reduced the runtime wie tried to refine it. There are too many possible (unconnected) solutions where the constraint is satisfied. Picture \ref{pic:rpl} shows one of it.
+\end{equation}
+As this constraint did not reduce the runtime we tried to refine it. There are too many possible (unconnected) solutions where the constraint is satisfied. Figure \ref{pic:rpl} shows one of it.
\begin{figure}
\centering
@@ -201,7 +201,7 @@ As this constraint did not reduced the runtime wie tried to refine it. There are
\label{pic:rpl}
\end{figure}
-This circumstane lead to the following constraint, that makes use of the gausian summ formula. The idea is still to limit the distance between the root node $v_{root}$ and all the members of $D$. In this advanced formulation we limit the sum of the distances to $\sum_{i_1}^|D*|{i}$. This constraint cuts of unconnected solutions that are valid using only the previous constraint \eqref{rpl}. But as our tests revealed this constraint did not generate a performance boost but even epanded the runtime(As it probably adds too much complexity to the model).
+This circumstance leads to the following constraint, that makes use of the Gaussian sum formula. The idea is still to limit the distance between the root node $v_{root}$ and all the members of $D$. In this advanced formulation we limit the sum of the distances to $\sum_{i}^{|D*|}{i}$. This constraint cuts off unconnected solutions that are valid using only the previous constraint \eqref{rpl}. But as our tests revealed this constraint did not generate a performance boost but even increased the runtime(As it probably adds too much complexity to the model).
(Maybe also mention that this constraint in isolation allows solutions which are forbidden using the previous one)
\subsubsection{Preventively adding separators}
diff --git a/Latex/preliminaries.tex b/Latex/preliminaries.tex
index 0e3f7b0..92ab896 100644
--- a/Latex/preliminaries.tex
+++ b/Latex/preliminaries.tex
@@ -3,20 +3,20 @@
Linear programming is a technique to minimize linear functions.
The following definition is based on the book \citep{fischetti2019introduction}\\
-A linear programm (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 an linear objective function that is minimized with respect to a set of linear inequalities. \\
\\
-Linear programms can be expressed as
+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 programms (ILPs) are linear programms with the additional restriction that all variables have to be integers: $x \in \mathbb{Z}^n$.
+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.
-Combinatorical optimisation problems can be modelled with ILPs. Every variable $x_i \in \{0,1\}$ denotes a possible decision to include item $i \in \{1,...,n\}$ in the solution.
+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: \[w \in N(v) \Leftrightarrow \exists (v,w) \in E\]
diff --git a/Latex/references.bib b/Latex/references.bib
index ea32410..39c0ed1 100644
--- a/Latex/references.bib
+++ b/Latex/references.bib
@@ -10,11 +10,11 @@ doi = {10.1007/s10878-017-0128-y}
}
@InProceedings{bomersbach,
-author="Bomersbach, Anna
-and Chiarandini, Marco
-and Vandin, Fabio",
-editor="Frith, Martin
-and Storm Pedersen, Christian N{\o}rgaard",
+author="Bomersbach, A.
+and Chiarandini, M.
+and Vandin, F.",
+editor="Frith, M.
+and Storm P., C. N{\o}rgaard",
title="An Efficient Branch and Cut Algorithm to Find Frequently Mutated Subnetworks in Cancer",
booktitle="Algorithms in Bioinformatics",
year="2016",
@@ -29,7 +29,7 @@ isbn="978-3-319-43681-4"
title = "Thinning out Steiner trees: a node based model for uniform edge costs",
abstract = "The Steiner tree problem is a challenging NP-hard problem. Many hard instances of this problem are publicly available, that are still unsolved by state-of-the-art branch-and-cut codes. A typical strategy to attack these instances is to enrich the polyhedral description of the problem, and/or to implement more and more sophisticated separation procedures and branching strategies. In this paper we investigate the opposite viewpoint, and try to make the solution method as simple as possible while working on the modeling side. Our working hypothesis is that the extreme hardness of some classes of instances mainly comes from over-modeling, and that some instances can become quite easy to solve when a simpler model is considered. In other words, we aim at “thinning out” the usual models for the sake of getting a more agile framework. In particular, we focus on a model that only involves node variables, which is rather appealing for the “uniform” cases where all edges have the same cost. In our computational study, we first show that this new model allows one to quickly produce very good (sometimes proven optimal) solutions for notoriously hard instances from the literature. In some cases, our approach takes just few seconds to prove optimality for instances never solved (even after days of computation) by the standard methods. Moreover, we report improved solutions for several SteinLib instances, including the (in)famous hypercube ones. We also demonstrate how to build a unified solver on top of the new node-based model and the previous state-of-the-art model (defined in the space of arc and node variables). The solver relies on local branching, initialization heuristics, preprocessing and local search procedures. A filtering mechanism is applied to automatically select the best algorithmic ingredients for each instance individually. The presented solver is the winner of the DIMACS Challenge on Steiner trees in most of the considered categories.",
keywords = "Exact computation, Mixed integer programming",
-author = "Matteo Fischetti and M. Leitner and Ivana Ljubic and Martin Luipersbeck and Michele Monaci and Max Resch and Domenico Salvagnin and Markus Sinnl",
+author = "M. Fischetti and M. Leitner and I. Ljubic and M. Luipersbeck and M. Monaci and M. Resch and D. Salvagnin and M. Sinnl",
year = "2017",
doi = "10.1007/s12532-016-0111-0",
language = "English",
@@ -41,22 +41,8 @@ publisher = "Springer Berlin Heidelberg",
number = "2",
}
-@InProceedings{number_v_sep,
-author="Gaspers, Serge
-and Mackenzie, Simon",
-editor="Mayr, Ernst W.",
-title="On the Number of Minimal Separators in Graphs",
-booktitle="Graph-Theoretic Concepts in Computer Science",
-year="2016",
-publisher="Springer Berlin Heidelberg",
-address="Berlin, Heidelberg",
-pages="116--121",
-abstract="We consider the largest number of minimal separators a graph on n vertices can have.",
-isbn="978-3-662-53174-7"
-}
-
@article{on_imposing_con,
-author = {Wang, Yiming and Buchanan, Austin and Butenko, Sergiy},
+author = {Wang, Y. and Buchanan, A. and Butenko, S.},
year = {2017},
month = {02},
pages = {},
@@ -73,7 +59,7 @@ doi = {10.1007/s10107-017-1117-8}
publisher={Independently Published}
}
@article{forrest,
-author = {Carvajal, Rodolfo and Constantino, Miguel and Goycoolea, Marcos and Vielma, Juan and Weintraub, Andres},
+author = {Carvajal, R. and Constantino, M. and Goycoolea, M. and Vielma, J. and Weintraub, A.},
year = {2013},
month = {08},
pages = {824-836},
@@ -83,7 +69,7 @@ journal = {Operations Research},
doi = {10.2307/23481799}
}
@article{fault_tolerant,
-author = {Buchanan, Austin and Sung, Je and Butenko, Sergiy and Pasiliao, Eduardo},
+author = {Buchanan, A. and Sung, J. and Butenko, S. and Pasiliao, E.},
year = {2015},
month = {02},
pages = {178-188},
@@ -93,7 +79,7 @@ journal = {INFORMS Journal on Computing},
doi = {10.1287/ijoc.2014.0619}
}
@book{ilp_np,
-author = {Garey, Michael R. and Johnson, David S.},
+author = {Garey, M. R. and Johnson, D. S.},
title = {Computers and Intractability; A Guide to the Theory of NP-Completeness},
year = {1990},
isbn = {0716710455},
@@ -101,9 +87,9 @@ publisher = {W. H. Freeman and Co.},
address = {USA}
}
@InProceedings{mtz,
-author="Fan, Neng
-and Watson, Jean-Paul",
-editor="Lin, Guohui",
+author="Fan, N.
+and Watson, J.-P.",
+editor="Lin, G.",
title="Solving the Connected Dominating Set Problem and Power Dominating Set Problem by Integer Programming",
booktitle="Combinatorial Optimization and Applications",
year="2012",
@@ -131,7 +117,7 @@ pages = {326–329},
numpages = {4}
}
@article{mtz_improv,
-author = {Desrochers, Martin and Laporte, Gilbert},
+author = {Desrochers, M. and Laporte, G.},
title = {Improvements and Extensions to the Miller-Tucker-Zemlin Subtour Elimination Constraints},
year = {1991},
issue_date = {February, 1991},
@@ -149,7 +135,7 @@ numpages = {10},
keywords = {subtour elimination constraints, vehicle routing problem, facets, lifting, traveling salesman problem}
}
@article{bio_netw,
-author = {Conn, Adam and Pedmale, Ullas and Chory, Joanne and Navlakha, Saket},
+author = {Conn, A. and Pedmale, U. and Chory, J. and Navlakha, S.},
year = {2017},
month = {07},
pages = {53-62.e3},
@@ -159,7 +145,7 @@ journal = {Cell Systems},
doi = {10.1016/j.cels.2017.06.017}
}
@article{bio_nutrient,
-author = {Posada, Juan and Sievänen, Risto and Messier, Christian and Perttunen, Jari and Nikinmaa, Eero and Lechowicz, Martin},
+author = {Posada, J. and Sievänen, R. and Messier, C. and Perttunen, J. and Nikinmaa, E. and Lechowicz, M.},
year = {2012},
month = {06},
pages = {731-41},
@@ -169,7 +155,7 @@ journal = {Annals of botany},
doi = {10.1093/aob/mcs106}
}
@article{bio_veinh,
-author = {Sack, Lawren and Scoffoni, Christine},
+author = {Sack, L. and Scoffoni, C.},
year = {2013},
month = {04},
pages = {},
@@ -179,7 +165,7 @@ journal = {The New phytologist},
doi = {10.1111/nph.12253}
}
@bachelorsthesis{myky,
- author={Hyunh, My Ky},
+ author={Hyunh, M. K.},
title={Solving Dominating Set Using Answer Set Programming},
school={Heinrich Heine University Düsseldorf},
year={2020},
@@ -187,9 +173,44 @@ doi = {10.1111/nph.12253}
}
@misc{klau,
title={Solving the Maximum-Weight Connected Subgraph Problem to Optimality},
- author={Mohammed El-Kebir and Gunnar W. Klau},
+ author={M. El-Kebir and G. W. Klau},
year={2014},
eprint={1409.5308},
archivePrefix={arXiv},
primaryClass={cs.DS}
}
+@BOOK{watertransport,
+ AUTHOR = {Nobel, P. S.},
+ TITLE = {Physicochemical and Environmental Plant Physiology},
+ PUBLISHER = {Elsevier},
+ YEAR = {2009},
+ EDITION ={4.}
+}
+@techreport{esa,
+ author = {D. Rehfeldt and H. Franz and T. Koch},
+ title = {Optimal Connected Subgraphs: Formulations and Algorithms},
+ institution = {ZIB},
+ address = {Takustr. 7, 14195 Berlin},
+ number = {20-23},
+ language = {eng},
+ urn = {urn:nbn:de:0297-zib-79094},
+ year = {2020}
+}
+@Inbook{esa_init,
+author="{\'A}lvarez-Miranda, E.
+and Ljubi{\'{c}}, I.
+and Mutzel, P.",
+editor="J{\"u}nger, M.
+and Reinelt, G.",
+title="The Maximum Weight Connected Subgraph Problem",
+bookTitle="Facets of Combinatorial Optimization: Festschrift for Martin Gr{\"o}tschel",
+year="2013",
+publisher="Springer Berlin Heidelberg",
+address="Berlin, Heidelberg",
+pages="245--270",
+abstract="The Maximum (Node-) Weight Connected Subgraph Problem (MWCS) searches for a connected subgraph with maximum total weight in a node-weighted (di)graph. In this work we introduce a new integer linear programming formulation built on node variables only, which uses new constraints based on node-separators. We theoretically compare its strength to previously used MIP models in the literature and study the connected subgraph polytope associated with our new formulation. In our computational study we compare branch-and-cut implementations of the new model with two models recently proposed in the literature: one of them using the transformation into the Prize-Collecting Steiner Tree problem, and the other one working on the space of node variables only. The obtained results indicate that the new formulation outperforms the previous ones in terms of the running time and in terms of the stability with respect to variations of node weights.",
+isbn="978-3-642-38189-8",
+doi="10.1007/978-3-642-38189-8_11",
+url="https://doi.org/10.1007/978-3-642-38189-8_11"
+}
+
diff --git a/Latex/results.tex b/Latex/results.tex
index 426b816..537a183 100644
--- a/Latex/results.tex
+++ b/Latex/results.tex
@@ -1,8 +1,5 @@
\section{Results}\raggedbottom
-(Ganz wichtig noch zu erwähnen, dass wir immer nur eine! optimale Lösung gesucht haben, da ILP das auch so macht!)
-
-
-This section shows our results of the runtime for the Minimum Connected rooted k-hop Dominating Set problem. We test the graphs that represent plant leafs from \citep{myky} as well as randomly generated graphs and grid graphs. At first we briefly describe the graphs from \citep{myky} and our other testgraphs. A more detailled description of the leaf graphs can be taken from \citep{myky}. All tests have been performed using a notebook with an Intel Core i7-4720HQ CPU @ 2.60GHz x 8 and 8 GB of RAM under Ubuntu 18.04.14 LTS.
+This section shows our results of the runtime for the Minimum Connected rooted k-hop Dominating Set problem. We test the graphs that represent plant leafs from \citep{myky} as well as randomly generated graphs and grid graphs. At first we briefly describe the graphs from \citep{myky} and our other test graphs. A more detailed description of the leaf graphs can be taken from \citep{myky}. All tests have been performed using a notebook with an Intel Core i7-4720HQ CPU @ 2.60GHz x 8 and 8 GB of RAM under Ubuntu 18.04.14 LTS and in all test cases we only looked for one single optimal solution.
As leaf graphs we use the instances \textit{small-leaf}, \textit{middle-leaf}, \textit{bigger-leaf}, \textit{maple} and \textit{asymmetric}. The instances \textit{small-leaf}, \textit{middle-leaf} and \textit{bigger-leaf} are similar in their structure. Each of the three graphs has the root at the bottom side and a symmetrical composition. They only differ in the number of nodes. The smallest graph \textit{small-leaf} has only 15 nodes while \textit{middle-leaf} has 62 nodes and \textit{bigger-leaf} has 71 nodes. While \textit{maple} represents a maple's leaf having 118 nodes, \textit{asymmetric} is inspired by an alocasia leaf. The peculiarity here is that it has the root in the middle of the leaf. It has 378 nodes.
@@ -18,8 +15,9 @@ First of all we introduce a table demonstrating different characteristics of the
Additionally the maximal, the average and the minimal node degree is shown. These parameters imply if a graph has at least one node with a much higher degree than the average or if the degrees are equally distributed.
\begin{table}[H]
+\centering
\begin{tabular}{l cccP{1.2cm}P{1.2cm}P{1.2cm}P{1.2cm}}
- name & |V| & |E| & densitiy & max. degree & avg degree & median degree & min degree \\
+ name & |V| & |E| & density & max. degree & avg degree & median degree & min degree \\
\hline
small-leaf & 15 & 30 & 0.29 & 6 & 4 & 4 & 1 \\
middle-leaf & 62 & 152 & 0.08 & 6 & 5 & 6 & 1\\
@@ -31,10 +29,11 @@ Additionally the maximal, the average and the minimal node degree is shown. Thes
\end{table}
We then continue with the runtime of our ILP-implementation using the leaf graphs as input.
-The following tables present the runtime in seconds as well as the number of constraints that were lazily added in the solution process. The last column shows the size of an optimal solution, i.e., the number of nodes that form a minumum dominating set.
-For the case that the solution process took more than 1000 seconds we state the upper bound and the lower bound that were determined within this time. The upper bound specifies the smallest solution that was found until the time was over. This means that an optimal solution will not be larger than the upper bound. In constrast the lower bound gives the smallest theoretical possible size of an optimal solution to that time. Let $U$ be an upper bound and $L$ be an lower bound. In the colum \textit{optimal} we used the denotion $[U,L]$.
+The following tables present the runtime in seconds as well as the number of constraints that were lazily added in the solution process. The last column shows the size of an optimal solution, i.e., the number of nodes that form a minimum dominating set.
+For the case that the solution process took more than 1000 seconds we state the upper bound and the lower bound that were determined within this time. The upper bound specifies the smallest solution that was found until the time was over. This means that an optimal solution will not be larger than the upper bound. In contrast the lower bound gives the smallest theoretical possible size of an optimal solution to that time. Let $U$ be an upper bound and $L$ be an lower bound. In the column \textit{optimal} we used the denotion $[U,L]$.
\begin{table}[H]
+\centering
\begin{tabular}{l cccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal \\
\hline
@@ -57,14 +56,15 @@ For the case that the solution process took more than 1000 seconds we state the
\caption[Minimum Connected rooted $k$-hop Dominating Set Results on the leaf graphs]{Minimum Connected rooted $k$-hop Dominating Set Results on the leaf graphs}
\end{table}
-With increasing parameter $k$ the runtime decreases significantly. Additionaly this table indicates a relation between the number of constraints that are added lazily and the runtime. Besides some outliers it seems like a high number of lazily added constraints implies a higher runtime. The more constraints that are added the more frequent unconnected integer solutions are found in the solution process. This effect occurs espacially on input instances that have many symmetrical solutions which are unconnected. If the input graph only has nodes that have a degree close to the average degree, then more likely this instance has many different symmetrical solutions. In such instances there is no node that is so valuable that it has to be included in the solution. If an unconnected integer solution is generated violated constraints are added to the model. After adding these constraints it most likely is cheaper to swap the nodes and use nodes where no violated constraints have been added yet than to use the same nodes and add those nodes, that the added constraints demand. On graphs where some nodes exist that have a significant higher degree than the average adding constraints more likely will not exclude them from a solution as they cover to many other vertices. This effect is roughly indicated by the number of lazily added constraints. If only a few constraints were added then there probably will not have been many options to swap valuable nodes without creating to many costs.
+With increasing parameter $k$ the runtime decreases significantly. Additionally this table indicates a relation between the number of constraints that are added lazily and the runtime. Besides some outliers it seems like a high number of lazily added constraints implies a higher runtime. The more constraints that are added the more frequent unconnected integer solutions are found in the solution process. This effect occurs especially on input instances that have many symmetrical solutions which are unconnected. If the input graph only has nodes that have a degree close to the average degree, then more likely this instance has many different symmetrical solutions. In such instances there is no node that is so valuable that it has to be included in the solution. If an unconnected integer solution is generated violated constraints are added to the model. After adding these constraints it most likely is cheaper to swap the nodes and use nodes where no violated constraints have been added yet than to use the same nodes and add those nodes, that the added constraints demand. On graphs where some nodes exist that have a significant higher degree than the average adding constraints more likely will not exclude them from a solution as they cover to many other vertices. This effect is roughly indicated by the number of lazily added constraints. If only a few constraints were added then there probably will not have been many options to swap valuable nodes without creating to many costs.
-With increasing size, i.e., number of nodes a graph has, the density of our graphs decreases. The density of the graph is another indicator that roughly implies the runtime \citep{fault_tolerant}. Especially on graphs with unequal distribution of node degrees. As with increasing size the density decreases on our graphs, the tests can not clearly indicate if the size is purely responsible for the runtime or if the density also has an influence. In the following we will test random generated graphs that have different size and for each size 10 different levels of density. On this graphs the densitiy clearly is the determing factor for the runtime.
+With increasing size, i.e., number of nodes a graph has, the density of our graphs decreases. The density of the graph is another indicator that roughly implies the runtime \citep{fault_tolerant}. Especially on graphs with unequal distribution of node degrees. As with increasing size the density decreases on our graphs, the tests can not clearly indicate if the size is purely responsible for the runtime or if the density also has an influence. In the following we will test random generated graphs that have different size and for each size 10 different levels of density. On this graphs the density clearly is the determining factor for the runtime.
The next table shows the characteristics of the random graphs.
\begin{table}[H]
+\centering
\begin{tabular}{l cccP{1.2cm}P{1.2cm}P{1.2cm}P{1.2cm}}
- name & |V| & |E| & densitiy & max. degree & avg. degree & median degree & min degree\\
+ name & |V| & |E| & density & max. degree & avg. degree & median degree & min degree\\
\hline
GNM\_ 50\_ 122 & 50 & 122 & 0.1 & 9 & 5 & 5 & 1 \\
GNM\_ 50\_ 245 & 50 & 245 & 0.2 & 15 & 10 & 9.5 & 6\\
@@ -110,159 +110,61 @@ The next table shows the characteristics of the random graphs.
\caption[The characteristics of the random graphs]{The characteristics of the random graphs}
\end{table}
-We have random graphs of four levels of size(|V| = 50; 100; 250; 500). For each of these levels we have ten levels of density(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0) to explore particulary its influence on the runtime.
+We have random graphs of four levels of size(|V| = 50; 100; 250; 500). For each of these levels we have ten levels of density(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0) to explore particularly its influence on the runtime. In the following table we only present the results for the density levels 0.1, 0.5 and 0.9 . The appendix contains the complete tables.
The results clearly show that, despite the larger size of the random graphs, the runtime is significantly shorter than on the leaf graphs. The density here seems to be a reasonable parameter that implies the runtime. On dense graphs few nodes are mandatory to form a dominating set. This allows to find an optimal solution faster.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
GNM\_ 50\_ 122 & 1 & 66 & 0.034878 & 11\\
- GNM\_ 50\_ 245 & 1 & 9 & 0.07 & 7\\
- GNM\_ 50\_ 368 & 1 & 0 & 0.013882 & 5 \\
- GNM\_ 50\_ 490 & 1 & 4 & 0.016478 & 4\\
- GNM\_ 50\_ 612 & 1 & 0 & 0.017783 & 4\\
- GNM\_ 50\_ 735 & 1 & 3 & 0.018471 & 3\\
- GNM\_ 50\_ 858 & 1 & 3 & 0.038161 & 3\\
- GNM\_ 50\_ 980 & 1 & 3 & 0.023549 & 3\\
- GNM\_ 50\_ 1102 & 1 & 3 & 0.019566 & 3\\
- GNM\_ 50\_ 1225 & 1 & 0 & 0.002396 & 1\\
- GNM\_ 100\_ 495 & 1 & 113 & 0.376731 & 14\\
- GNM\_ 100\_ 990 & 1 & 17 & 0.488522 & 8\\
- GNM\_ 100\_ 1485 & 1 & 7 & 0.396982 & 6\\
- GNM\_ 100\_ 1980 & 1 & 0 & 0.315584 & 5\\
- GNM\_ 100\_ 2475 & 1 & 0 & 0.045136 & 4\\
- GNM\_ 100\_ 2970 & 1 & 0 & 0.013737 & 3\\
- GNM\_ 100\_ 3465 & 1 & 0 & 0.010702 & 3\\
- GNM\_ 100\_ 3960 & 1 & 0 & 0.007955 & 2\\
- GNM\_ 100\_ 4455 & 1 & 0 & 0.00505 & 2\\
- GNM\_ 100\_ 4950 & 1 & 0 & 0.00535 & 1\\
- GNM\_ 250\_ 3112 & 1 & 0 & 1017.303471 & [17;15]\\
- GNM\_ 250\_ 6225 & 1 & 0 & 900.64 & 10 \\
- GNM\_ 250\_ 9338 & 1 & 0 & 29.67 & 7\\
- GNM\_ 250\_ 12450 & 1 & 0 & 46.78 & 6\\
- GNM\_ 250\_ 15562 & 1 & 0 & 12.29 & 5\\
- GNM\_ 250\_ 18675 & 1 & 0 & 0.97 & 4\\
- GNM\_ 250\_ 21788 & 1 & 3 & 0.415836 & 3\\
- GNM\_ 250\_ 24900 & 1 & 0 & 0.040482 & 3\\
- GNM\_ 250\_ 28012 & 1 & 0 & 0.024473 & 2\\
- GNM\_ 250\_ 31125 & 1 & 0 & 0.017227 & 1\\
- GNM\_ 500\_ 12475 & 1 & 42 & 1004.920676 & [21;13]\\
- GNM\_ 500\_ 24950 & 1 & 0 & 1051.277153 & [12;8]\\
- GNM\_ 500\_ 37425 & 1 & 0 & 9.89 & 4\\
- GNM\_ 500\_ 49900 & 1 & 0 & 1017.23594 & [6;5]\\
- GNM\_ 500\_ 62375 & 1 & 0 & 178.495614 & 5\\
- GNM\_ 500\_ 74850 & 1 & 0 & 9.753998 & 4\\
- GNM\_ 500\_ 87325 & 1 & 0 & 21.368156 & 4\\
- GNM\_ 500\_ 99800 & 1 & 0 & 0.286309 & 3\\
- GNM\_ 500\_ 112275 & 1 & 0 & 0.189313 & 2\\
- GNM\_ 500\_ 124750 & 1 & 0 & 0.11 & 1\\
- \end{tabular}
- \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]
- \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\_ 122 & 3 & 0 & 2 & 0.01651\\
+ GNM\_ 50\_ 612 & 1 & 0 & 0.017783 & 4\\
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\_ 612 & 3 & 0 & 1 & 0.002541\\
+ GNM\_ 50\_ 1102 & 1 & 3 & 0.019566 & 3\\
GNM\_ 50\_ 1102 & 2 & 0 & 1 & 0.012025\\
- GNM\_ 50\_ 1225 & 2 & 0 & 1 & 0.001756\\
+ GNM\_ 50\_ 1102 & 3 & 0 & 1 & 0.012196\\
+ GNM\_ 100\_ 495 & 1 & 113 & 0.376731 & 14\\
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\_ 495 & 3 & 0 & 1 & 0.026969\\
+ GNM\_ 100\_ 2475 & 1 & 0 & 0.045136 & 4\\
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\_ 2475 & 3 & 0 & 1 & 0.006448\\
+ GNM\_ 100\_ 4455 & 1 & 0 & 0.00505 & 2\\
GNM\_ 100\_ 4455 & 2 & 0 & 1 & 0.003927\\
- GNM\_ 100\_ 4950 & 2 & 0 & 1 & 0.003468\\
+ GNM\_ 100\_ 4455 & 3 & 0 & 1 & 0.004094\\
+ GNM\_ 250\_ 3112 & 1 & 0 & 1017.303471 & [17;15]\\
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\_ 3112 & 3 & 14 & 1 & 0.141794\\
+ GNM\_ 250\_ 15562 & 1 & 0 & 12.29 & 5\\
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]
- \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 & 1 & 0 & 0.024473 & 2\\
+ GNM\_ 250\_ 28012 & 2 & 0 & 1 & 0.018999\\
GNM\_ 250\_ 28012 & 3 & 0 & 1 & 0.023179\\
- GNM\_ 250\_ 31125 & 3 & 0 & 1 & 0.020695\\
+ GNM\_ 500\_ 12475 & 1 & 42 & 1004.920676 & [21;13]\\
+ GNM\_ 500\_ 12475 & 2 & 0 & 2 & 1.123904\\
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 & 1 & 0 & 178.495614 & 5\\
+ GNM\_ 500\_ 62375 & 2 & 0 & 1 & 0.29011\\
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 & 1 & 0 & 0.189313 & 2\\
+ GNM\_ 500\_ 112275 & 2 & 0 & 1 & 0.148031\\
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}
+ \caption[Minimum Connected rooted $k$-hop Dominating Set Results on the random graphs]{Minimum Connected rooted $k$-hop Dominating Set Results on the random graphs}
\end{table}
-We also tested another class of graphs on their runtime. The structure of our leaf graphs is similar in the manner that all have a fixed neighborhood of 6 vertices, all are planar and almost all nodes have the same degree. Many grid graphs also have all these characteristics. This is why we tested our implementation also on grid graphs. Here we tested graphs that are quadratic as well as graphs that are more oblong. Especially on qudratic graphs the same behavior like on the leaf graphs has occured. Here also comparatively many constraints were added lazily. Which indicates that here also many unconnected integer solutions were created. It seems like the ``gridness'' of a graph a the crucial factor that pushs the runtime over a reasonable extent. The gridness can be defined as the combination of the three described properties from the beginning. On grid graphs the ASP-version also performs much better than the ILP-Version.
+We also tested another class of graphs on their runtime. The structure of our leaf graphs is similar in the manner that all have a fixed neighborhood of 6 vertices, all are planar and almost all nodes have the same degree. Many grid graphs also have all these characteristics. This is why we tested our implementation also on grid graphs. Here we tested graphs that are quadratic as well as graphs that are more oblong. Especially on quadratic graphs the same behavior like on the leaf graphs has occurred. Here also comparatively many constraints were added lazily. Which indicates that here also many unconnected integer solutions were created. It seems like the ``gridness'' of a graph a the crucial factor that pushs the runtime over a reasonable extent. The gridness can be defined as the combination of the three described properties from the beginning. On grid graphs the ASP-version also performs much better than the ILP-Version.
Here also a short overview about the characteristics of the grid graphs.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
- name & |V| & |E| & densitiy & max. degree & avg degree & median degree & min degree\\
+ name & |V| & |E| & density & max. degree & avg degree & median degree & min degree\\
\hline
GRID\_ 6\_ 4 & 24 & 38 & 0.14 & 4 & 3 & 3 & 2\\
GRID\_ 8\_ 8 & 64 & 112 & 0.06 & 4 & 4 & 4 & 2\\
@@ -274,6 +176,7 @@ Here also a short overview about the characteristics of the grid graphs.
\end{table}
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
@@ -303,6 +206,7 @@ In the method section (refer at this place) we introduced the MTZ constraints (a
The next table shows the runtime of three graphs using the MTZ constraints.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & runtime(s) & optimal\\
\hline
@@ -313,12 +217,13 @@ The next table shows the runtime of three graphs using the MTZ constraints.
\caption[Minimum Connected rooted $k$-hop Dominating Set Results using the MTZ constraints]{Minimum Connected rooted $k$-hop Dominating Set Results using the MTZ constraints}
\end{table}
-The version using the vertex sepearator is in all testes cases many times faster. The version using the MTZ constraints seems not to be a reasonable alternative.
+The version using the vertex separator is in all testes cases many times faster. The version using the MTZ constraints seems not to be a reasonable alternative.
-Now we study the case when some of the vertex separator constraints are preadded to the model. We preadded for all combinations $c_v$ of a vertex $v$ and its neighborhood $N(v) $the vertex separators that seperate $c_v$ and the root vertex $v_r$. As the following table reveals this generates a significant speedup to the runtime. However the bigger leaf instance can still not be solved optimal under 1000 seconds. In all test cases the ILP-version with preadeed separators performed better than the ASP-version. Still many separator constraints needed to be added lazily. If these constraints can be identified in advance this could generate another speedup.
+Now we study the case when some of the vertex separator constraints are preadded to the model. We preadded for all combinations $c_v$ of a vertex $v$ and its neighborhood $N(v) $the vertex separators that separate $c_v$ and the root vertex $v_r$. As the following table reveals this generates a significant speedup to the runtime. However the bigger leaf instance can still not be solved optimal under 1000 seconds. In all test cases the ILP-version with preadded separators performed better than the ASP-version. Still many separator constraints needed to be added lazily. If these constraints can be identified in advance this could generate another speedup.
At this point preadding the described separators itself does not improve the ILP-implementation in a manner that the runtime is satisfying.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
@@ -334,9 +239,10 @@ At this point preadding the described separators itself does not improve the ILP
At last we present tables that show the effect of the additional constraints (referenz) introduced in the method section(ref) on some graphs.
-The first table shows the effect of the \textit{intermediate node constraint}(ref) from \citep{fischetti_steiner_t}. To recap this constraint demands that every vertex that is part of the dominating set needs at least two neighbors which are also members of the dominating set. Roughly speaking every node of the dominating set(except for the root) needs to be an intermediate node. This constraint reduces the runtime drasticly. Howewer in most cases including this constraint adds nodes to the solution that would not be included without. For example the instances \textit{middle-leaf} and \textit{bigger-leaf} have one extra node in the optimal solution when this constraint is included.
+The first table shows the effect of the \textit{intermediate node constraint}(ref) from \citep{fischetti_steiner_t}. To recap this constraint demands that every vertex that is part of the dominating set needs at least two neighbors which are also members of the dominating set. Roughly speaking every node of the dominating set(except for the root) needs to be an intermediate node. This constraint reduces the runtime drastically. However in most cases including this constraint adds nodes to the solution that would not be included without. For example the instances \textit{middle-leaf} and \textit{bigger-leaf} have one extra node in the optimal solution when this constraint is included.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
@@ -352,22 +258,24 @@ The first table shows the effect of the \textit{intermediate node constraint}(re
The next table shows the results using the naive constraint to reduce the path length from the root to members of the dominating set. It does not reduce the runtime but even increases it. For the cases were we stopped the solution process after a fixed time span the upper bounds and lower bounds are worse than without this constraint. However in some cases this constraint reduces the number of lazily added constraints which is an indicator that the room of possible unconnected solutions was reduced. But this effect did not reduce the runtime. Probably this constraint added complexity to the model which increased the runtime instead.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
small-leaf & 2 & 9 & 0.008948 & 3 \\
middle-leaf & 2 & 109 & 10.936048 & 14\\
bigger-leaf & 2 & 67 & 23.457956 & 15 \\
- maple & 118 & 2 & 5804 & 1011.766479 & [26,20] \\
+ maple & 2 & 5804 & 1011.766479 & [26,20] \\
asymmetric & 2 & 17391 & 1114.582689 & [190,81] \\
\end{tabular}
\caption[Minimum Connected rooted $k$-hop Dominating Set Results with SPL constraint]{Minimum Connected rooted $k$-hop Dominating Set Results with SPL constraint}
\end{table}
-The additional constraint that uses the gaussian sumformula even performed drastically worse. The runtime increased significantly as this constraints adds a high degree of complexity to the model.
+The additional constraint that uses the Gaussian sum formula even performed drastically worse. The runtime increased significantly as this constraints adds a high degree of complexity to the model.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
@@ -378,9 +286,10 @@ The additional constraint that uses the gaussian sumformula even performed drast
\caption[Minimum Connected rooted $k$-hop Dominating Set Results with GAUS constraint]{Minimum Connected rooted $k$-hop Dominating Set Results with GAUS constraint}
\end{table}
-When using both constraints in conjunction the constraint with the gaussian sumformula dominates the runtime.
+When using both constraints in conjunction the constraint with the Gaussian sum formula dominates the runtime.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
@@ -396,11 +305,12 @@ As we compared our ILP-version to the ASP-version from \citep{myky} the followin
We start with our leaf graphs. This table clearly shows that the ASP-version performs much better on these graphs. As for example for the \textit{middle-leaf} instance with parameter $k=1$ the ASP-version finds a solution in 154 seconds, after 1100 seconds the ILP-version does not find a solution(ref).
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccc}
name & k & \# lazily added constraints & runtime(s) & optimal\\
\hline
small-leaf & 1 & 9 & 0.008 & 6\\
- small-leaf & 2 & 1 & 4 & 0.009 & 3\\
+ small-leaf & 2 & 4 & 0.009 & 3\\
small-leaf & 3 & 0 & 0.009 & 2\\
middle-leaf & 1 & 4945 & 153.605 & 22\\
middle-leaf & 2 & 2043 & 0.597 & 14\\
@@ -409,7 +319,7 @@ We start with our leaf graphs. This table clearly shows that the ASP-version per
bigger-leaf & 2 & 377 & 1.735 & 15 \\
bigger-leaf & 3 & 1266 & 0.069 & 11\\
maple & 1 & 194321 & 1129.807776 & [41,31]\\
- maple & 2 & 118 & 9621 & 1008.548 & [26,24]\\
+ maple & 2 & 9621 & 1008.548 & [26,24]\\
maple & 3 & 8029 & 1006.839 & [21,20]\\
asymmetric & 1 & 34255 & 1011.016 & [164, 29]\\
asymmetric & 2 & 2706 & 1009.839 & [102,20]\\
@@ -422,152 +332,54 @@ We start with our leaf graphs. This table clearly shows that the ASP-version per
We continue with the runtime of the ASP-version on random graphs. This tables clearly indicate that the ILP-version performs better on random graphs.
\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]
- \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]
- \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 & 1 & 0.055 & 4\\
+ GNM\_ 50\_ 612 & 2 & 1 & 0.038\\
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 & 1 & 0.052 & 3\\
+ GNM\_ 50\_ 1102 & 2 & 1 & 0.052\\
GNM\_ 50\_ 1102 & 3 & 1 & 0.051\\
- GNM\_ 50\_ 1225 & 3 & 1 & 0.053\\
+ GNM\_ 100\_ 495 & 1 & 32.451 & 14\\
+ GNM\_ 100\_ 495 & 2 & 4 & 0.084\\
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 & 1 & 0.655 & 4\\
+ GNM\_ 100\_ 2475 & 2 & 1 & 0.151\\
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 & 1 & 0.253 & 2 \\
+ GNM\_ 100\_ 4455 & 2 & 1 & 0.220\\
GNM\_ 100\_ 4455 & 3 & 1 & 0.227\\
- GNM\_ 100\_ 4950 & 3 & 1 & 0.223\\
+ GNM\_ 250\_ 3112 & 1 & 1017.204 & [23;9]\\
+ GNM\_ 250\_ 3112 & 2 & 2 & 0.521\\
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 & 1 & 1008.099 & [5;4]\\
+ GNM\_ 250\_ 15562 & 2 & 1 & 0.972\\
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 & 1 & 3.400 & 2\\
+ GNM\_ 250\_ 28012 & 2 & 1 & 1.453\\
GNM\_ 250\_ 28012 & 3 & 1 & 1.489\\
- GNM\_ 250\_ 31125 & 3 & 1 & 1.510\\
+ GNM\_ 500\_ 12475 & 1 & 1016.396 & [29;7]\\
+ GNM\_ 500\_ 12475 & 2 & 2 & 2.314\\
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 & 1 & 1006.141 & [6;4]\\
+ GNM\_ 500\_ 62375 & 2 & 1 & 4.218\\
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 & 1 & 8.705 & 2\\
+ GNM\_ 500\_ 112275 & 2 & 1 & 6.268\\
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}
+ \caption[Minimum Connected rooted $k$-hop Dominating Set Results on the random graphs using ASP]{Minimum Connected rooted $k$-hop Dominating Set Results on the random graphs using ASP}
\end{table}
On the other hand the ASP-version performs better on the grid graphs. This is as we expected.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccccccccc}
name & k & runtime(s) & optimal\\
\hline
@@ -591,9 +403,10 @@ On the other hand the ASP-version performs better on the grid graphs. This is as
\end{table}
At very last we want to have a deeper look into one particular aspect. During the solution process upper and lower bounds are determined. Most of the time the ILP-version is capable of finding a solid upper bound quickly. The vast majority of the time needed to find an optimal solution is spent on closing the gap to the lower bound. To illustrate this the next table shows after what time an upper bound that is 20\%, 10\%, 5\% and 0\% different from an optimal solution is found.
-In the cases were the ASP-version performs better it also founds a proper upper bound faster. In the one case where the ILP-version performs better it finds an appriopriate upper bound faster.
+In the cases were the ASP-version performs better it also founds a proper upper bound faster. In the one case where the ILP-version performs better it finds an appropriate upper bound faster.
\begin{table}[H]
+\centering
\begin{tabular}{l ccccccP{1cm}P{1cm}cc}
name & type & k & 20\% & 10\% & 5\% & 0\% & time to close the gap & \# lazily added constraints & runtime(s) & optimal\\
\hline
--
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