\begin{table}[htbp]
\centering
\caption{Performance of different objective reduction algorithms on Non-redundant test problems, corresponding to $\mathcal{N}_{R}$ and $\theta=0.997$. The numbers in the table indicate the frequency of success in identifying the $\mathcal{F}_{\mathcal{T}}$, out of 20 runs. The dashes (-) denote cases which were not tried due to huge computational time}
\centering
\footnotesize{
\begin{tabularx}{\columnwidth}{cc cccc}
\toprule
Probl. & \multicolumn{2}{c}{Proposed approaches} & \multicolumn{2}{c}{DRP} \\
D(M) & NL-MVU-PCA & L-PCA & Greedy & Exact \\
\midrule
1(05) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) \\
1(15) & 01 ($12.4\pm1.6$) & 15 ($14.6\pm0.8$) & 20 ($15.0\pm0.0$) & 20 ($15.0\pm0.0$) \\
1(25) & 00 ($17.4\pm2.3$) & 15 ($23.2\pm1.7$) & 02 ($23.0\pm0.9$) & -- \\
2(05) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) \\
2(15) & 06 ($13.7\pm1.1$) & 19 ($14.9\pm0.2$) & 17 ($14.8\pm0.3$) & 16 ($14.8\pm0.4$) \\
2(25) & 00 ($19.0\pm1.8$) & 06 ($23.3\pm1.7$) & 01 ($21.8\pm1.2$) & -- \\
3(05) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) \\
3(15) & 05 ($13.6\pm1.1$) & 20 ($15.0\pm0.0$) & 19 ($14.9\pm0.2$) & 18 ($14.9\pm0.3$) \\
3(25) & 00 ($18.8\pm1.7$) & 06 ($23.1\pm2.0$) & 03 ($23.2\pm1.2$) & -- \\
4(05) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) \\
4(15) & 20 ($15.0\pm0.0$) & 20 ($15.0\pm0.0$) & 00 ($11.9\pm0.9$) & 01 ($12.2\pm0.9$) \\
4(25) & 20 ($25.0\pm0.0$) & 20 ($25.0\pm0.0$) & 00 ($11.2\pm0.5$) & -- \\
7(05) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) & 20 ($05.0\pm0.0$) \\
7(15) & 20 ($15.0\pm0.0$) & 20 ($15.0\pm0.0$) & 00 ($12.8\pm0.8$) & 00 ($11.8\pm0.7$) \\
7(25) & 20 ($25.0\pm0.0$) & 20 ($25.0\pm0.0$) & 00 ($11.3\pm0.5$) & -- \\
\bottomrule
\end{tabularx}
}
\label{tab:pc-nonredundant}
\end{table}

\begin{table}[htbp]
\centering
\caption{Performance of different objective reduction algorithms on Redundant test problems, corresponding to $\mathcal{N}_{R}$ and $\theta=0.997$. The numbers in the table indicate the frequency of success in identifying the $\mathcal{F}_{\mathcal{T}}$, out of 20 runs. The dashes (-) denote cases which were not tried due to huge computational time}
\centering
\footnotesize{
\begin{tabularx}{\columnwidth}{cc cccc}
\toprule
Probl. & \multicolumn{2}{c}{Proposed approaches} & \multicolumn{2}{c}{DRP} \\
D(M) & NL-MVU-PCA & L-PCA & Greedy & Exact \\
\midrule
5(2,05) & 0/0 ($3.90\pm0.31$) & 0/0 ($05.00\pm0.00$) & 0/0 ($05.00\pm0.00$) & 0/0 ($05.00\pm0.00$) \\
5(2,10) & 0/0 ($4.55\pm0.69$) & 0/0 ($10.00\pm0.00$) & 0/0 ($10.00\pm0.00$) & 0/0 ($10.00\pm0.00$) \\
5(2,20) & 0/0 ($5.25\pm1.07$) & 0/0 ($13.45\pm1.64$) & 0/0 ($19.85\pm0.37$) & 0/0 ($19.85\pm0.37$) \\
5(2,30) & 0/0 ($6.50\pm1.36$) & 0/0 ($13.15\pm3.05$) & 0/0 ($26.80\pm1.51$) & 0/0 ($26.75\pm1.59$) \\
5(2,50) & 0/0 ($7.65\pm1.39$) & 0/0 ($09.85\pm2.87$) & 0/0 ($34.25\pm3.26$) & -- \\
5(3,05) & 4/4 ($3.90\pm0.55$) & 0/0 ($05.00\pm0.00$) & 0/0 ($05.00\pm0.00$) & 0/0 ($05.00\pm0.00$) \\
5(3,10) & 2/2 ($5.25\pm1.16$) & 0/0 ($09.65\pm0.81$) & 0/0 ($10.00\pm0.00$) & 0/0 ($10.00\pm0.00$) \\
5(3,20) & 0/2 ($6.40\pm0.82$) & 0/0 ($09.75\pm1.41$) & 0/0 ($19.60\pm0.60$) & 0/0 ($19.60\pm0.60$) \\
5(5,10) & 9/9 ($5.55\pm0.51$) & 0/0 ($07.85\pm0.99$) & 0/0 ($10.00\pm0.00$) & 0/0 ($10.00\pm0.00$) \\
5(5,20) & 0/0 ($7.05\pm0.94$) & 0/0 ($07.70\pm0.98$) & 0/0 ($17.40\pm1.05$) & 0/0 ($17.35\pm1.04$) \\
5(7,10)&14/12 ($7.30\pm0.47$) & 1/0 ($08.35\pm0.59$) & 0/0 ($10.00\pm0.00$) & 0/0 ($10.00\pm0.00$) \\
5(7,20) & 1/1 ($8.50\pm0.83$) & 0/0 ($09.35\pm0.88$) & 0/0 ($15.70\pm1.78$) & 0/0 ($15.50\pm1.82$) \\
 W3(05)&20/20 ($2.00\pm0.00$) & 0/0 ($03.00\pm0.00$) & 0/0 ($04.90\pm0.31$) & 0/0 ($04.00\pm0.00$) \\
 W3(15) & 2/2 ($4.70\pm1.49$) & 5/0 ($04.05\pm1.61$) & 0/0 ($04.95\pm0.22$) & 0/0 ($04.00\pm0.00$) \\
 W3(25) & 3/3 ($4.10\pm1.37$) & 6/0 ($03.90\pm1.48$) & 0/0 ($05.00\pm0.00$) & 0/0 ($04.00\pm0.00$) \\
\bottomrule
\end{tabularx}
}
\label{tab:pc-redundant}
\end{table}