## 3 Results
### Results on Phase 2. We implemented Steps 4 and 5 in Phase 2 as follows. **Step 4. ** In this step, we solve the MILP \(\mathcal{M}(x,y,g;\mathcal{C}_{1},\mathcal{C}_{2})\) formulated based on the ANN \(\mathcal{N}\) obtained in Phase 1. To solve an MILP in Step 4, we use CPLEX version 12.10. In our experiment, we choose a target value \(y^{*}\in[\underline{a},\overline{a}]\) and fix or bound some descriptors in our feature vector as follows:  
- Fix variable \(\theta\) that represents the polymer parameter \(\theta(H)\) to be each integer in \(\{-2,0,2\}\);  
- Set \(d_{\max}\) to be each of 3 and 4;  
- Fix \(n^{*}\) to be some four integers in \(\{15,19,20,25,30\}\) for \(\theta\in\{-2,0\}\) and \(\{15,19,20,22,25\}\) for \(\theta=2\);  
- Choose three integers from \([7,16]\) and fix cs\({}^{*}\) to be each of the three integers;  
- Fix ch\({}^{*}\) to be each of the four integers in \([2,5]\). Based on the above setting, we generated 12 instances for each \(n^{*}\). We set \(\varepsilon=0.02\) in Step 4. Tables 3-8 show the results of Step 4 for \(d_{\max}=3\) and 4, respectively, where we denote the following:  
- \(y^{*}_{\pi}\): a target value in \([\underline{a},\overline{a}]\) for a property \(\pi\);  
- \(n^{*}\): a specified number of vertices in \([\underline{u},\overline{n}]\);  
- \(|F^{*}|/\#\): #1 means the number of MILP instances in Step 4 (where #1 = 12), and \(|F^{*}|\) means the size of set \(F^{*}\) of vectors \(x^{*}\) generated from all feasible instances among the #1 MILP instances in Step 4;  
- IP-time: the average time (sec.) to solve one of the #1 MILP instances to find a set \(F^{*}\) of vectors \(x^{*}\). Figure 9(a)-c illustrate some rank-2 chemical graphs \(G^{*}\) with \(\theta(G^{*})=-2\) constructed from the vector \(g^{*}\) obtained by solving the MILP in Step 4. Figure 10(a)-c illustrate some rank-2 chemical graphs \(G^{*}\) with \(\theta(G^{*})=0\) constructed from the vector \(g^{*}\) obtained by solving the MILP in Step 4. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \(\pi\) & \(y^{*}\) & \(n^{*}\) & \(|F^{*}|/\#1\) & IP-Time & \(\#G^{*}\) & G-Time \\ \hline Kow & 5 & 15 & 11/12 & 31.84 & 100 & 413.8 \\ Kow & 5 & 20 & 12/12 & 69.65 & 12 & \(>\)1 h \\ Kow & 5 & 25 & 12/12 & 144.20 & 11 & \(>\)1 h \\ Kow & 5 & 30 & 12/12 & 352.01 & 12 & \(>\)1 h \\ Mp & 150 & 15 & 9/12 & 20.68 & 100 & 947.4 \\ Mp & 150 & 20 & 11/12 & 73.73 & 11 & \(>\)1 h \\ Mp & 150 & 25 & 9/12 & 140.09 & 9 & \(>\)1 h \\ Mp & 150 & 30 & 12/12 & 304.04 & 12 & \(>\)1 h \\ Br & 250 & 15 & 7/12 & 28.51 & 100 & 232.7 \\ Br & 250 & 19 & 11/12 & 82.01 & 11 & \(>\)1 h \\ Br & 250 & 22 & 12/12 & 150.55 & 12 & \(>\)1 h \\ Br & 250 & 25 & 12/12 & 239.84 & 12 & \(>\)1 h \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of Steps 4 and 5 with \(d_{\max}=4\) and \(\theta=-2\). \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \(\pi\) & \(y^{*}\) & \(n^{*}\) & \(|F^{*}|/\#1\) & IP-Time & \(\#G^{*}\) & G-Time \\ \hline Kow & 5 & 15 & 11/12 & 31.84 & 100 & 413.8 \\ Kow & 5 & 20 & 12/12 & 69.65 & 12 & \(>\)1 h \\ Kow & 5 & 25 & 12/12 & 144.20 & 11 & \(>\)1 h \\ Kow & 5 & 30 & 12/12 & 352.01 & 12 & \(>\)1 h \\ Mp & 150 & 15 & 9/12 & 20.68 & 100 & 947.4 \\ Mp & 150 & 20 & 11/12 & 73.73 & 11 & \(>\)1 h \\ Mp & 150 & 25 & 9/12 & 140.09 & 9 & \(>\)1 h \\ Mp & 150 & 30 & 12/12 & 304.04 & 12 & \(>\)1 h \\ Br & 250 & 15 & 7/12 & 28.51 & 100 & 232.7 \\ Br & 250 & 19 & 11/12 & 82.01 & 11 & \(>\)1 h \\ Br & 250 & 22 & 12/12 & 150.55 & 12 & \(>\)1 h \\ Br & 250 & 25 & 12/12 & 239.84 & 12 & \(>\)1 h \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of Steps 4 and 5 with \(d_{\max}=3\) and \(\theta=-2\).