## 5 Optimization Method  
This section presents an optimization method that decreases the computational effort of solving the MILP problem presented in Section 4.2. The method consists of using a reduced set of candidate routing paths instead of the complete set of paths \(\mathcal{P}\) and solving model (2)-(24) assuming these paths. It is a heuristic method based on the price-and-branch approach [19], which uses the CG technique, and it is called the price-and-branch algorithm (PBA) in the remainder of the paper.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Objective** & minimize \(z=\sum_{v\in\mathcal{V}^{\rm HP}}z_{v}\) & (2) \\ \hline \multicolumn{3}{l}{**Constraints**} \\ \multicolumn{3}{l}{placement of DU/CU} \\ \hline  & \(\sum_{v\in\mathcal{V}^{\rm HP}}u_{v\sigma}=1\), & \(\forall c\in\mathcal{C}\), & (3) \\  & \(u_{v\sigma}^{\rm DU}=u_{\sigma\sigma}\), & \(\forall c\in\mathcal{C}\), \(d\in\mathcal{D}(c),v\in\mathcal{V}^{\rm PP}\), & (4) \\  & \(u_{v\sigma}\leq z_{v}\), & \(\forall v\in\mathcal{V}^{\rm PP}\), \(c\in\mathcal{C}\), & (5) \\  & \(\sum_{v\in\mathcal{V}^{\rm HP}}u_{v\sigma}^{\rm CU-URLLC}=1\), & \(\forall s\in\mathcal{S}^{\rm URLLC}\), & (6) \\  & \(u_{v\sigma}^{\rm DU}=u_{v\sigma}^{\rm CU-URLLC}\), & \(\forall s\in\mathcal{S}^{\rm URLLC}\), \(d\in\mathcal{D}(s),v\in\mathcal{V}^{\rm PP}\), & (7) \\  & \(u_{v\sigma}^{\rm CU-URLLC}\leq z_{v}\), & \(\forall v\in\mathcal{V}^{\rm PP}\), \(s\in\mathcal{S}^{\rm URLLC}\), & (8) \\  & \(\sum_{v\in\mathcal{P}}P^{\rm DU}(d)\cdot u_{d\sigma}^{\rm DU}+\sum_{s\in \mathcal{S}^{\rm URLLC},d\in\mathcal{D}(s)}\mathcal{P}^{\rm CU}(d)\cdot u_{d \sigma}^{\rm CU}\leq C(v)\), & \(\forall v\in\mathcal{V}^{\rm PP}\), & (9) \\ \hline \multicolumn{3}{l}{path selection for flows} \\ \hline \([\lambda_{d}^{\rm FH}]\) & \(\sum_{v\in\mathcal{P}(d,\sigma)}x_{dfp}-1=0\), & \(\forall d\in\mathcal{D},f=\{\rm FH\}\), & (10) \\  & \(u_{d\sigma}^{\rm DU}-\sum_{p\in\mathcal{P}(d,\sigma)}x_{dfp}=0\), & \(\forall d\in\mathcal{D},f=\{\rm FH\}\), \(v\in\mathcal{V}^{\rm PP}\), & (11) \\  & \(u_{d\sigma}^{\rm DU}-\sum_{p\in\mathcal{P}(d,\sigma)}x_{dfp}=0\), & \(\forall s\in\mathcal{S}^{\rm eMBB},d\in\mathcal{D}(s),f=\{\rm MH\},v\in \mathcal{V}^{\rm PP}\), & (12) \\  & \(u_{d\sigma}^{\rm DU}-\sum_{p\in\mathcal{P}(d,\sigma)}x_{dfp}-u_{d\sigma}^{ \rm DU}=0\), & \(\forall s\in\mathcal{S}^{\rm URLLC},d\in\mathcal{D}(s),f=\{\rm MH\},v\in \mathcal{V}^{\rm PP}\), & (13) \\  & \(u_{d\sigma}^{\rm DU}-\sum_{p\in\mathcal{P}(d,\sigma)}x_{dfp}-u_{d\sigma}^{ \rm DU}=0\), & \(\forall s\in\mathcal{S}^{\rm URLLC},d\in\mathcal{D}(s),f=\{\rm MH\},v\in \mathcal{V}^{\rm PP}\), & (14) \\ \hline \multicolumn{3}{l}{usage of network links} \\ \hline \([\kappa_{e}\geq 0]\) & \(H(e)-\sum_{l\in\mathcal{D}}\sum_{f\in\mathcal{F}}\sum_{v\in\mathcal{P}(d, \sigma)}H(d,f)\cdot x_{dfp}\geq 0\), & \(\forall e\in\mathcal{E}\), & (15) \\  & \(y_{d\varepsilon}-\sum_{p\in\mathcal{P}(d,\sigma)}x_{dfp}=0\), & \(\forall d\in\mathcal{D},f\in\mathcal{F}\), \(e\in\mathcal{E}\), & (16) \\ \hline \multicolumn{3}{l}{latency of flows} \\ \hline  & \(\sum_{(d,f)\in L}(\bar{d},\bar{f},e)\cdot y_{d\varepsilon}+\bar{L}(d,f,e)\cdot(y_ {d\varepsilon}-1)\leq w_{d\varepsilon}^{\rm HP}\), & \(\forall d\in\mathcal{D},f\in\mathcal{F}\),\(e\in\mathcal{E}^{S}\) & (17) \\  & \(\sum_{(\bar{d},\bar{f})\in L(\bar{d},f,e)}y_{d\bar{f}e}\leq|\mathcal{L}(d,f,e,t)| \cdot y_{d\varepsilon}^{\rm LP}\), & \(\forall d\in\mathcal{D},f\in\mathcal{F}\),\(e\in\mathcal{E}^{S},t\in\mathcal{T}(d,f,e)\), & (18) \\  & \(L(t)\cdot\left(y_{d\varepsilon}^{\rm LP}+y_{d\varepsilon}-1\right)\leq w_{d \varepsilon}^{\rm LP}\), & \(\forall d\in\mathcal{D},f\in\mathcal{F}\),\(e\in\mathcal{E}^{S}\),\(t\in\mathcal{T}(d,f,e)\), & (19) \\  & \(w_{d\varepsilon}^{\rm HP}+w_{d\varepsilon}^{\rm LP}=w_{d\varepsilon}^{\rm HP}\), & \(\forall d\in\mathcal{D},f\in\mathcal{F}\),\(e\in\mathcal{E}^{S}\), & (20) \\  & \(y_{d\varepsilon}