## 4 Proposed Method
### Policy Optimization with Separation-of-Duty Constraints
#### 4.2.2 Construction of t-m MEAR Constraints from k-n SOAR Constraints
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(|\)**U**\(|\) & \(|\)**U**\(|\) & \(|\)**U**\(AV\) & \(|\)**\(OAV\)** & \(|\)**\(RC_{max}\)** & \(|\)**\(RL_{max}\)** & \(|\)**\(P_{X\text{-Stoller}}\)** & \(|\)**\(P_{PEO\_VR\&SOD}\)** \\ \hline
1000 & 100 & 20 & 20 & 30 & 5 & 27 & 26.31 \\
1000 & 100 & 25 & 25 & 30 & 5 & 25.67 & 25.67 \\
1000 & 100 & 30 & 30 & 30 & 5 & 26.67 & 25.11 \\
1000 & 100 & 35 & 35 & 30 & 5 & 26.67 & 25.37 \\
1000 & 100 & 40 & 40 & 30 & 5 & 27.53 & 26.89 \\
1000 & 100 & 25 & 25 & 20 & 5 & 17.67 & 17.53 \\
1000 & 100 & 25 & 25 & 40 & 5 & 36 & 35.31 \\
1000 & 100 & 25 & 25 & 50 & 5 & 43 & 42.83 \\
1000 & 100 & 25 & 25 & 30 & 4 & 25.67 & 25.39 \\
1000 & 100 & 25 & 25 & 30 & 3 & 24.67 & 24.64 \\
1000 & 100 & 25 & 25 & 30 & 2 & 24 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 19: Comparison of rules with Xu-Stoller for different parameters. Figure 7: Comparison of time with Xu-Stoller for different \(|RL|\) values. These figures show that, for PEO_VR&SOD, the execution time tends to grow linearly and does not obviously vary as the number of attributes values, the value of \(|RC|\), and/or the value of \(|RL|\) vary. However, for Xu-Stoller, Figure 5 demonstrates that the execution time first increases exponentially and then decreases gradually with an increasing number of attribute values. Specifically, the time increases remarkably from around 33 s to around 8000 s when the number of attribute values varies from 20 to 30 and then decreases gradually to around 7000 s when the value is close to 40. Further, the time required in our method is below 100 s for any case, while Xu-Stoller requires significantly more time with an increasing number of attribute values. PEO_VR&SOD runs faster than Xu-Stoller because the former eliminates the redundant attributes from each candidate rule, as shown in Lines 22-29 of Algorithm 2, while the latter takes more time in the generalization step while eliminating attribute expressions with constraints. Figures 6 and 7 demonstrate that the execution time irregularly fluctuates up and down with increasing values of \(|RC|\) and \(|RL|\). Detailed analyses are not discussed in this paper, similar to the analyses in [27]. Thus, PEO_VR&SOD outperforms Xu-Stoller on these datasets. In the above evaluations, we implement the experiments for Xu-Stoller and PEO_VR&SOD with the same experimental setups: considering not only the varying number of users and objects as well as that of their attributes, but also number of rules used to construct the datasets and the maximum length of any mining rule. However, we do not need to consider all of these setups for VisMAP on the same datasets. Therefore, VisMAP will outperform Xu-Stoller as well as PEO_VR&SOD for execution time on the same datasets. ### Performance Evaluation of Enforcement of SOD Constraints

To evaluate the effectiveness of PEO_VR&SOD, we study the performance of enforcing SOD constraints in the policy optimization stage, which can be converted into a study on the effects of enforcing MEAR constraints. We also employed real datasets (as shown in Table 17) that were used to construct an ABAC system in the policy mining stage. Further, the SOD constraints were synthetically constructed similar to the examples in this paper, such as the 3-5 SOD constraints. Moreover, the Rel-SAT model counter [37] is used for constructing the SOAR constraints. As shown by Definition 8 and Statement 2, the verification of the MEAR constraints is primarily affected by the number of users, as well as that of the rules. Therefore, we implemented experiments in datasets where the number of users varies from 20 to 50 with a step of 10, and the number of rules varies from 5 to 20 with a step of 5. The execution time for verification with a varying number of users and rules is shown in Figure 8.

Figure 8 shows that the execution time tends to grow linearly and does not obviously vary with an increasing number of users for each specific policy configuration.