## 2 System Model and Problem Statement
### Power Consumption Model and Problem Formulation
The power expenditure model for the mmWave links in \(\mathcal{E}_{wl}\) follows [25] (see Equations (7)-(10) therein); specifically, the power consumed on link \(e\in\mathcal{E}_{wl}\) is given by  
\[P_{e}^{(mmW)}=N_{RF}^{(mmW)}\Big{(}\chi_{e}P_{e}^{(mmW,i)}+\Delta_{e}^{(mmW)}F( \ell_{e})\Big{)}, \tag{5}\]  
where \(N_{RF}^{(mmW)}\) is the number of Radio Frequency (RF) chains in the link, \(P_{e}^{(mmW,i)}\) is the idle power of the link's transmitter, \(\ell_{e}\triangleq\sum_{q\in\mathcal{Q}}\sum_{e^{\prime}\in\mathcal{E}^{(q)} }\theta_{e}^{\prime,q}/b_{e}\) is a load-dependent variable (where \(b_{e}\) is the utilized bandwidth of link \(e\)), \(\Delta_{e}^{(mmW)}\) is a slope parameter depending on the power electronics used in the link's transmitter, and \(F(\cdot)\) is a **piecewise-linear** function accounting for the nonlinear dependence between achieved throughput and power expenditure. Finally, \(\chi_{e}\) is a Boolean indicator variable for whether link \(e\) is **actually** used to serve any traffic; if not, the link's transceiver is turned off. The power expenditure for the AN links in \(\mathcal{E}_{AN}\) is similarly modeled as follows: the power expended by a gNB BS \(a\in\mathcal{A}\) is  
\[P_{a}^{(gNB)}=N_{RF}^{(gNB)}\cdot\left(\mu_{a}P_{a}^{(gNB,i)}+\Delta_{a}^{(gNB )}\sum_{j\in\mathcal{S}^{(a)}}x_{j,a}p_{a}^{(RB)}N_{a,j}^{(RB)}\right), \tag{6}\]  
where \(\mu_{a}\) is the Boolean indicator for whether BS \(a\) actually serves any UEs, and \(N_{RF}^{(gNB)}\), \(P_{a}^{(SNB,i)}\), and \(\Delta_{a}^{(gNB)}\) have the same semantics as in (5). For an SC BS, it similarly holds  
\[P_{a}^{(SC)}=N_{RF}^{(SC)}\cdot\left(\mu_{a}P_{a}^{(SC,i)}+\Delta_{a}^{(SC)} \sum_{j\in\mathcal{S}^{(a)}}x_{j,a}p_{a}^{(RB)}N_{a,j}^{(RB)}\right). \tag{7}\]  
We use the clever trick of [25] to convert the activation/power saving constraints into a set of linear constraints by introducing auxiliary Boolean variables \(\chi_{e}\), for \(e\in\mathcal{E}_{wl}\), and \(\
u_{a}\), for \(a\in\mathcal{A}\)  
\[\chi_{e}+C_{2}\sigma_{e}\geq 1,\ \forall\ e\in\mathcal{E}_{wl},\]  
\[1-C_{2}\chi_{e}\leq\sum_{q\in\mathcal{Q}}\sum_{e^{\prime}\in\mathcal{E}^{(p)_{ q}}}\theta_{e}^{\prime,q}\leq C_{2}(1-\sigma_{e}),\ \forall\ e\in\mathcal{E}_{wl}, \tag{8}\]  
\[1-C_{1}\
u_{a}\leq\sum_{j\in\mathcal{J}}x_{j,a}\leq C_{2}(1-\
u_{a}),\ \mu_{a}+C_{1}\
u_{a}\geq 1,\ \forall a\in \mathcal{A}.\]In addition to the basic self-consistency conditions  
\[ \begin{split}&\phi_{s_{q_{j}},dum1,q_{j}}=\phi_{j,dum2,q_{j}}=1, \quad\forall\,j\in\mathcal{J},\\ &\phi_{g,f_{dum1,q_{j}}}=0,\quad\forall\,j\in\mathcal{J},\;\forall \,\hat{y}\in\mathcal{V}_{c}\setminus\{s_{q_{j}}\},\\ &\phi_{g,f_{dum2,q_{j}}}=0,\quad\forall\,j\in\mathcal{J},\;\forall \,\hat{y}\in\mathcal{V}_{c}\setminus\{j\},\end{split}  \tag{9}\]  
and  
\[ \begin{split}&\phi_{j,f,q}=0,\quad\forall\,j\in\mathcal{J},\; \forall\,q\in\mathcal{Q},\;\forall\,f\in\mathcal{F}\setminus\{f_{dum2}\},\\ &\phi_{g,f,q}=0,\quad\forall\,q\in\mathcal{Q},\;\forall\,f\in \mathcal{F}\setminus\mathcal{R}_{p_{q}},\;\forall\,\hat{y}\in\mathcal{V}_{c} \cup\mathcal{J},\\ &\theta_{a,j}^{e_{q_{j}}^{\prime},q_{j}}=x_{j,a},\quad\forall\,j \in\mathcal{J},\;\forall\,a\in\mathcal{A},\end{split}  \tag{10}\]  
following from the definitions, where \(e_{q_{j}}^{\prime}\) is the \"last\" virtual edge in \(\mathcal{E}^{(\rho_{q})}\), i.e., \(t\Big{(}e_{q_{j}}^{\prime}\Big{)}=1\), we also impose the constraints:  
\[ \begin{split}&\sum_{y\in\mathcal{V}_{c}}\phi_{y,f,q}=1,\quad \forall\,q\in\mathcal{Q},\;\forall\,f\in\mathcal{R}_{q}\setminus\mathcal{F}_ {dum},\\ &\sum_{a\in\mathcal{A}^{(j)}}x_{j,a}=1,\quad\forall\,j\in\mathcal{ J},\\ & x_{j,b}=0,\quad\forall\,b\
ot\in\mathcal{A}^{(j)},\end{split}  \tag{11}\]  
which require that each requested VNF must be deployed at a node and that each UE must properly attach to exactly one of its allowable BSs (i.e., those BSs which can allocate a sufficient number of Resource Blocks to serve its requested traffic). The constraint  
\[