## 6 Numerical Results
### Performance of PBA vs. MILP  
In this subsection, we compare the PBA with the MILP method. In Table 5, we present detailed results of the objective function values (\(z\)) and computation times (\(T\)) obtained in network MESH-20, assuming different numerologies (\(\mu\) \begin{table}
\begin{tabular}{c c c} \hline
**Flow** & **Uplink** & **Downlink** \\ \hline Fronthaul & 21.624 & 22.204 \\ Midhaul & 3.024 & 4.016 \\ Backhaul & 3 & 4 \\ \hline \end{tabular}
\end{table} 
Table 4: Maximum bit rates [Gbit/s] of Xhaul data flows per an RU. Figure 5: The impact of the number of candidate paths (\(k\)) on the objective function value and computation times in network MESH-20. numbers of RUs (\(R\)), \(k=5\), and \(\gamma=0.2\). We show MILP optimality gaps (\(\Delta^{\text{MILP}}\)) and relative differences, denoted as \(\Delta^{\text{z}}\) and \(\Delta^{\text{T}}\), between the values of \(z^{\text{MILP}}\) and \(z^{\text{PBA}}\) and \(T^{\text{MILP}}\) and \(T^{\text{PBA}}\) obtained for MILP and the PBA, respectively. Also, we report the number of columns (variables) in the MILP model using the entire set of paths (cols\({}^{\text{MILP}}\)) and after running the CG procedure (cols\({}^{\text{PBA}}\)), the initial heuristic solution (\(z^{\text{init}}\)), and the runtimes of both the CG procedure (\(T^{\text{CG}}\)) and the CPLEX solving the MILP model with the generated columns (\(T^{\text{BB}}\)). In Table 5, we can see that the MILP solutions are optimal (\(\Delta^{\text{MILP}}=0\%\)) only for \(\mu=1\), and for \(R\leq 30\) RUs, when \(\mu\in\{2,3\}\). In these cases, the results of \(z\) achieved with the PBA are the same; hence, they are also optimal. In the remaining network scenarios, the PBA significantly outperforms the MILP approach, for which the optimality gap is high (even above 60%) after the 1 h of computation. Namely, the results of \(z\) are much lower for the PBA than for MILP, where the relative difference (\(\Delta^{\text{z}}\)) is about 40-50%. Such a difference can be explained by the complexity of solving the MILP model, in particular the number of problem variables, which is some tens of times smaller in the PBA (cols\({}^{\text{PBA}}\)) than in MILP (cols\({}^{\text{MILP}}\)). Indeed, there is a large difference in the CPLEX solving times (\(T^{\text{MILP}}\) and \(T^{\text{BB}}\)) in both methods, which comes from the use of a limited number of paths (generated using CG) in the PBA instead of the complete set of paths in MILP. The main computation effort of the PBA is related to the CG procedure, where \(T^{\text{CG}}\) exceeds hundreds of seconds for larger \(R\). Even though, the overall PBA runtimes (\(T^{\text{PBA}}\)) compared to MILP (\(T^{\text{MILP}}\)) can be reduced from a few to tens of times (\(\Delta^{\text{T}}\)). In Figures 6-8, we extend the analysis of the MILP and PBA performance to a larger set of network scenarios. Namely, we compare the objective function values (\(z\) in Figure 6) and the computation times (\(T\) in Figure 7) and report relative differences for these two metrics (\(\Delta^{\text{z}}\) and \(\Delta^{\text{T}}\) in Figure 8) for different numbers of RUs (\(R\)) in networks MESH-20 and MESH-38. The results are obtained and averaged over \(\mu\in\{1,2,3\}\), \(\gamma\in\{0.2,0.3\}\), \(k\in\{3,5\}\), and \(\rho=1.5\). In addition, the average optimality gaps of the MILP method (\(\Delta^{\text{MILP}}\), denoted as b