## 4 Experiments
\begin{table}
\begin{tabular}{c|c c c c c c} \hline  & \multicolumn{5}{c}{**Problem**} \\ \hline \multicolumn{2}{c}{**Fragments**} & 22,448 & 89,718 & 298,194 & 730,201 & 2,278,504 \\ \multicolumn{2}{c}{**Contigs Obtained**} & 82 & 114 & 163 & 224 & 510 \\ \multicolumn{2}{c}{**Total Number of Bases**} & 19,343 & 84,659 & 240,314 & 552,408 & 1,321,969 \\ \hline \multirow{10}{*}{**Contigs**} & **Contigs Found** & 74 & 101 & 149 & 204 & 480 \\  & **\%** & 90.2 & 88.6 & 91.4 & 91.1 & 94.1 \\ \cline{2-6}  & **Bases Found** & 15,822 & 70,250 & 206,826 & 463,582 & 1,102,152 \\ \cline{2-6}  & **\%** & 81.8 & 83 & 86.1 & 83.9 & 83.4 \\ \cline{2-6}  & **N50** & 360 & 1614 & 3326 & 4937 & 4499 \\ \cline{2-6}  & **Average Length** & 213.8 & 695.5 & 1388.10 & 2272.50 & 2296.20 \\ \cline{2-6}  & **Minimum Length** & 54 & 58 & 54 & 62 & 53 \\ \cline{2-6}  & **Maximum Length** & 882 & 7261 & 10,943 & 22,745 & 24,396 \\ \hline \multirow{10}{*}{**Contigs**} & **Contigs not Found** & 8 & 13 & 14 & 20 & 30 \\  & **\%** & 9.8 & 11.4 & 8.6 & 8.9 & 5.9 \\ \cline{1-1} \cline{2-6}  & **Bases not Found** & 3521 & 14,409 & 33,488 & 88,826 & 219,817 \\ \cline{1-1} \cline{2-6}  & **\%** & 18.2 & 17 & 13.9 & 16.1 & 16.6 \\ \cline{1-1} \cline{2-6}  & **Not Found** & **N50** & 758 & 2308 & 3954 & 7258 & 11,798 \\ \cline{1-1} \cline{2-6}  & **Average Length** & 440.1 & 1108.40 & 2392.00 & 4441.30 & 7327.20 \\ \cline{1-1} \cline{2-6}  & **Minimum Length** & 95 & 95 & 95 & 181 & 255 \\ \cline{1-1} \cline{2-6}  & **Maximum Length** & 826 & 3703 & 6809 & 9542 & 23,614 \\ \hline \end{tabular}
\end{table} 
Table 6: Algorithm 1, \(F_{1}\) objective function, all overlaps. \begin{table}
\begin{tabular}{c|c c c c c} \hline  & \multicolumn{5}{c}{**Problem**} \\ \hline \multicolumn{2}{c}{**Fragments**} & 22,448 & 89,718 & 298,194 & 730,201 & 2,278,504 \\ \multicolumn{2}{c}{**Contigs Obtained**} & 82 & 114 & 163 & 224 & 510 \\ \multicolumn{2}{c}{**Total Number of Bases**} & 19,028 & 84,584 & 240,280 & 552,509 & 1,331,757 \\ \hline \multirow{10}{*}{**Contigs**} & **Contigs Found** & 74 & 102 & 146 & 189 & 450 \\  & **\%** & 90.2 & 89.5 & 89.6 & 84.4 & 88.2 \\ \cline{1-1}  & **Bases Found** & 15,515 & 70,843 & 189,927 & 369,001 & 931,343 \\ \cline{1-1} \cline{2-6}  & **\%** & 81.5 & 83.8 & 79 & 66.8 & 69.9 \\ \cline{1-1} \cline{2-6}  & **N50** & 356 & 1527 & 3184 & 4298 & 4201 \\ \cline{1-1}  & **Average Length** & 209.7 & 694.5 & 1300.90 & 1952.40 & 2069.70 \\ \cline{1-1}  & **Minimum Length** & 54 & 58 & 54 & 59 & 53 \\ \cline{1-1}  & **Maximum Length** & 882 & 7261 & 10,943 & 21,185 & 16,135 \\ \hline \end{tabular}
\end{table} 
Table 7: Algorithm 1, \(F_{2}\) objective function, all overlaps. From the DNA fragment assembly point of view, we are interested in large _contigs_ with no errors. The most important elements to be computed are the number of bases and the _contigs_ that were found in the complete genome, as well as the mean length of each _contig_. When we compare the results obtained by Algorithm 1 for \(F_{1}\) and \(F_{2}\) using all overlaps, we find that \(F_{1}\) is a little bit better than \(F_{2}\), except in the number of bases found for problem 89,718. In terms of multi objective function optimization, the solution obtained for \(F_{1}\) is not enough to dominate the solution obtained by \(F_{2}\), but since it is better in most of the test, we can say that it is almost dominant. In the other experiments, we find a similar situation, where \(F_{1}\) is almost dominant with respect to \(F_{2}\). For Experiment 2, we excluded transitive overlaps and used Algorithm 1 for \(F_{1}\) and \(F_{2}\). The results are given in Tables 5, 8 and 9. In general, we found that \(F_{1}\) is better than \(F_{2}\) without dominating it. Comparing the results where transitive overlaps are excluded (Tables 8 and 9 or Table 5) to those where they are not (Tables 6 and 7 or Table 5), we can see that the results are similar for \(F_{1}\), hence it is not clear whether it is better to exclude or not to exclude transitive overlaps.