## 3 LDCP Problem
### Notation  
The NGFI network is represented by a directed and connected graph, denoted as \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), in which \(\mathcal{V}\) and \(\mathcal{E}\) are the sets of network nodes and links, respectively. Let \(\mathcal{V}^{R}\), \(\mathcal{V}^{P}\), \(\mathcal{V}^{DC}\), and \(\mathcal{V}^{S}\) denote the sets of RU, PP, DC, and switching nodes. These sets are disjoint and their sum constitutes the set of all nodes (\(\mathcal{V}\)). Subgraph \(\mathcal{G}^{S}=(\mathcal{V}^{S},\mathcal{E}^{S})\) represents the packet transport network, where \(\mathcal{E}^{S}\) is the set of links between the switching nodes (\(\mathcal{E}^{S}\subset\mathcal{E}\)). Let \(\mathcal{E}^{Sout}\) denote the set of output links of the switches (\(\mathcal{E}^{Sout}\subset\mathcal{E}\)). The RU, PP, and DC nodes from sets \(\mathcal{V}^{R}\), \(\mathcal{V}^{P}\), and \(\mathcal{V}^{DC}\) are connected with some nodes from set \(\mathcal{V}^{S}\) (i.e., with the transport network). Let \(K(e)\) denote the capacity of link \(e\in\mathcal{E}\) and let \(\rho(v)\) be the processing capacity of PP node \(v\in\mathcal{V}^{P}\). Let \(L^{P}(e)\) be the propagation delay in link \(e\). Let \(L^{SF}(e)\) be the store-and-forward delay of the switching node that is the origin node of link \(e\in\mathcal{E}^{Sout}\). We assume that \(L^{SF}(e)=0\) if link \(e\) is not originated in a switch. The set of clusters is denoted as \(\mathcal{C}\). Each cluster \(c\in\mathcal{C}\) represents a subset of RUs (\(c\subset\mathcal{V}^{R}\)), such that their DUs must be placed and processed together in the same PP node to facilitate multi-cell coordination (see Section 2.1). All RUs belong to some clusters and the clusters are disjoint, i.e., \(c_{1}\cap c_{2}=\emptyset,\forall c_{1},c_{2}\in\mathcal{C}\). Let \(\mathcal{D}\) be the set of demands. Demand \(d\in\mathcal{D}\) is identified with an RU node, and it represents a couple of traffic flows to be realized in the network. The flows are the following:  
1. a fronthaul flow--between the RU node and the PP node in which the DU entity is placed;
2. a midhaul flow--between the PP node in which the DU entity is located and a different PP node in which the CU entity is placed. Note that if the DU and CU are located in the same PP node for a given demand, then the MH flow is not present in the network for this demand;
3. a backhaul flow--between the PP node in which the CU entity is located and a DC node. Let \(\mathcal{F}\) denote the set of types of traffic flows, namely \(\mathcal{F}=\{\mathrm{FH,MH,BH}\}\). Let \(\rho^{D}\) and \(\rho^{C}\) be the processing requirements (loads) of DU and CU entities, respectively. We assume that for each RU there are two associated demands to be realized in the network, namely an uplink demand and a downlink demand. The uplink demand is realized from RU towards DU, CU, and DC, while the downlink demand has an opposite direction, namely from DC towards CU, DU, and RU. The set of uplink demands is denoted as \(\mathcal{D}^{U}\), and the set of downlink demands is denoted as \(\mathcal{D}^{D}\). These two sets are disjoint, have the same cardinality, and together they form set \(\mathcal{D}\), namely, \(\mathcal{D}=\mathcal{D}^{U}\cup\mathcal{D}^{D}\). The cluster comprising the RU of demand \(d\in\mathcal{D}\) is denoted as \(\mathcal{C}(d)\). Let \(\mathcal{Q}^{HEP}(d,f)\) and \(\mathcal{Q}^{LP}(d,f)\) denote the sets of demand-flow pairs \((\bar{d},\bar{f})\), where \(\bar{d}\in\mathcal{D}\) and \(\bar{f}\in\mathcal{F}\). Set \(\mathcal{Q}^{HEP}(d,f)\) comprises the pairs of either equal or higher priority than flow \(f\) of demand \(d\). Set \(\mathcal{Q}^{LP}(d,f)\) comprises the pairs of a lower priority than flow \(f\) of demand \(d\). For instance, if \(f\) is an MH flow, then \(\mathcal{Q}^{HEP}(d,f)\) will contain all the FH flows of all demands and the MH flows of all the demands except for \(d\). Concurrently, \(\mathcal{Q}^{LP}(d,f)\) will contain all the BH flows of all demands.