## 3 LDCP Problem
### MILP Formulation
--RUs clustering--it assures that the DU processing for all RUs belonging to a cluster is performed in the same PP node; in particular, \(\forall c\in\mathcal{C}\), the following constraint is imposed:  
\[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]  
--PP node assignment for DU processing--it assures that the DU processing of demands is performed in the PP nodes that have been assigned to the clusters containing the RUs of these demands; in particular, \(\forall d\in\mathcal{D},c=C(d),v\in\mathcal{V}^{p}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]  
--PP node selection for CU processing--it assures that single PP nodes are assigned for the purpose of CU processing for particular demands; in particular, \(\forall d\in\mathcal{D}\), the following constraint is imposed:  
\[\sum_{v\in\mathcal{V}^{p}}u_{dv}^{C}=1, \tag{4}\]  
--FH and BH flows--it assures that for fronthaul and backhaul flows there is a connection established, respectively, between the RU node and a PP node (in case of FH) and between a PP node and the DC node (in case of BH); in particular, \(\forall d\in\mathcal{D},f\in\{\mathrm{FH},\mathrm{BH}\}\), we have  
\[\sum_{i\in\mathcal{V}^{src}(d,f)}\sum_{j\in\mathcal{V}^{dst}(d,f)}x_{dfij}=1, \tag{5}\]  
--MH flow--assures, for each uplink and downlink demand \(d\), that there is either a MH flow established between a pair of PP nodes (if DU and CU and located in different PP nodes) or such a flow is not realized in the network (if DU and CU are placed in the same PP node); in particular, we have  
\[\sum_{j\in\mathcal{V}^{dst}(d,f)}x_{dfij}+u_{dv}^{CD} =u_{dv}^{D}\text{ for }d\in\mathcal{D}^{U},f=\{\mathrm{MH}\},v\in \mathcal{V}^{src}(d,f), \tag{6}\] \[\sum_{j\in\mathcal{V}^{dst}(d,f)}x_{dfij}+u_{dv}^{CD} =u_{dv}^{C}\text{ for }d\in\mathcal{D}^{D},f=\{\mathrm{MH}\},v\in \mathcal{V}^{src}(d,f),\] (7) \[\sum_{i\in\mathcal{V}^{src}(d,f)}x_{dfip}+u_{dv}^{CD} =u_{dv}^{C}\text{ for }d\in\mathcal{D}^{U},f=\{\mathrm{MH}\},v\in \mathcal{V}^{dst}(d,f),\] (8) \[\sum_{i\in\mathcal{V}^{src}(d,f)}x_{dfip}+u_{dv}^{CD} =u_{dv}^{D}\text{ for }d\in\mathcal{D}^{D},f=\{\mathrm{MH}\},v\in \mathcal{V}^{dst}(d,f), \tag{9}\]  
--Termination of FH and BH flows in the PP nodes selected--it assures that the fronthaul and backhaul flows of demands are terminated in the appropriate PP nodes, namely in which the DU and CU entities of the demands are placed; in particular, the following constraints are imposed:  
\[x_{dfiv} =u_{dv}^{D}\text{ for }d\in\mathcal{D}^{U},f=\{\mathrm{FH}\},i\in \mathcal{V}^{src}(d,f),v\in\mathcal{V}^{dest}(d,f) \tag{10}\] \[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]0 (11) \[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]1 (12) \[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]2  
--Activation of PP nodes--it assures that the PP nodes are active when there are DU/CU entities placed in these nodes; in particular, \(\forall v\in\mathcal{V}^{p},d\in\mathcal{D}\), the following constraints are imposed:  
\[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]3 \[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]4--Capacity of PP nodes--it assures that the overall DU and CU processing load in PP nodes does not exceed the capacity of these nodes; in particular, \(\forall v\in\mathcal{V}^{P}\), the following constraint is imposed:  
\[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]5  
--Capacity of links--it assures that the volume of traffic in network links is not greater than the capacity of links; in particular, \(\forall e\in\mathcal{E}\), the following constraint is imposed:  
\[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]6  
--Utilization of links--it allows to determine whether flows are carried over particular links; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F},e\in\mathcal{E}\), the following constraint is imposed:  
\[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]7  
--Interfering of flows--it allows to determine whether two different flows use the same switch output link; namely, \(x_{dfffe}\) is equal to 1 if and only if both flow \(f\) of demand \(d\) and flow \(\tilde{f}\) of demand \(\tilde{d}\) use link \(e\); in particular, \(\forall e\in\mathcal{E}^{Sout}\), and \(d,\tilde{d}\in\mathcal{D},f,\tilde{f}\in\mathcal{F}\), except for \(d=\tilde{d},f=\tilde{f}\), the following constraints are imposed:  
\[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]8 \[\sum_{v\in\mathcal{V}^{p}}y_{cv}=1, \tag{2}\]9 (20) \[u_{dv}^{D}=y_{cv}, \tag{3}\]0 (21) \[u_{dv}^{D}=y_{cv}, \tag{3}\]1  
--Dynamic latencies of flows because of the flows of equal/higher priority--it estimates worst-case latencies of flows in the output links of switches caused by either equal- or higher-priority flows; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F},e\in\mathcal{E}^{Sout}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]2  
--Dynamic latencies of flows because of the flows of lower priority--it estimates worst-case latencies of flows in the output links of switches caused by lower-priority flows; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F},(\tilde{d},\tilde{f})\in\mathcal{Q}^{ LP}(d,f),e\in\mathcal{E}^{Sout}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]3  
--Dynamic latencies of flows--it estimates worst-case latencies of flows produced in the output links of switches; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F},e\in\mathcal{E}^{Sout}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]4  
--Static latencies of flows--it estimates the latencies of flows produced in a network link as the sum of link propagation delay, store-and-forward delay produced in the origin node of the link (if the node is a switch) and burst transmission delay; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F},e\in\mathcal{E}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]5Latencies of flows--it estimates the latencies of flows as the sum of static and dynamic latencies produced in the network links over which the flows are routed; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]6  
--Maximum latencies of flows--it assures that the latency levels of fronthaul, midhaul, and backhaul flows are within allowable limits; in particular, \(\forall d\in\mathcal{D},f\in\mathcal{F}\), the following constraint is imposed:  
\[u_{dv}^{D}=y_{cv}, \tag{3}\]7  
The LDCP problem is \(\mathcal{NP}\)-complete.