## 2 CAP in Cellular Mobile Networks  
### The Minimum Span Frequency Assignment Problem (MS-FAP)  
The Minimum Span Frequency Assignment Problem (MS-FAP) is a CAP model in which the resulting channel assignment satisfies the demand of each cell, adheres to the channel spacing constraints between cells, and minimizes the system's bandwidth usage. This involves minimizing the gap between the maximum and minimum channel assignments in the entire system [54; 55].  
The conventional variables and nomenclature for the MS-FAP problem are given by [56], where:  
1. The network capacity is determined by \(M\) different non-negative channels, all of which are accessible to \(N\) different cells.
2. The demand vector is given by \(\mathbf{D}=[D_{1},\cdots,D_{i},\cdots,D_{N}]_{1\times N}\), where it contains the number of channels requested by the \(i\)-th cell.
3. The electromagnetic compatibility matrix \(\mathbf{C}=[C_{ij}]_{N\times N}\) contains the channel spacing constraints between the \(i\)-th and \(j\)-th cells. These constraints can be classified and interpreted as co-channel (CCC), adjacent (ACC), or co-site (CSC) limits [56].
4. The channel assignment matrix \(\mathbf{F}=[F_{ik}]_{N\times M}\) represents the assigned channel number \(F_{ik}\) to the \(i\)-th cell. These channels are chosen from a list of positive integers denoted by \(\{1,2,\cdots,M\}\) or assigned as zero if no channel was allocated.  
Using the variables and nomenclature previously shown, the MS-FAP is formulated, and the resulting optimization problem is given by:  
\[\min\left\{\max_{\forall i,k}\{F_{ik}\}\right\} \tag{1}\] \[\text{s.t.}\Big{|}F_{ik}-F_{jl}\Big{|}\geq C_{ij}, \forall k,l\in\{1,\cdots,M\}\] (2) \[\forall i\
eq j,\forall k\
eq l,\forall F_{ik},F_{jl}\in\mathbb{N}\] \[\text{s.t.}\sum_{k=1}^{M}\{1-\delta[F_{ik}]\}=D_{i},\forall i\in \{1,\cdots,N\} \tag{3}\]  
where \(\delta[x]\) denotes the Delta-Kronecker function, which returns 1 if \(x=0\) and 0 if \(x\
eq 0\). In this formulation, the goal of reducing network bandwidth utilization is represented by (1), compliance with electromagnetic limitations is represented by (2), and fulfillment of network demand is represented by (3).  
To clarify the nomenclature, we observe two cells being solved using the MS-FAP in Figure 1, where the electromagnetic compatibility matrix is given by  
\[\mathbf{C}=\left[\begin{array}{cc}2&1\\ 1&2\end{array}\right], \tag{4}\]  
the demand vector is \(\mathbf{D}=[3,2]\), and the non-zero frequencies are \(F_{1}=\{F_{11},F_{12},F_{13}\}\) and \(F_{2}=\{F_{21},F_{22}\}\) for cells one and two, respectively. For example, the assigned channels \(F_{1}=\{2,4,6\}\) and \(F_{2}=\{1,3\}\) are feasible solutions considering the electromagnetic compatibility and demand constraints. However, the MS-FAP aims to minimize the overall span (i.e., the network's bandwidth). Therefore, the only valid solution in this example is \(F_{1}=\{1,3,5\}\) and \(F_{2}=\{2,4\}\), where \(\min\{\max_{\forall i,k}\{F_{ik}\}\}=5\) represents the minimum of the maximum frequencies that satisfy all constraints. As a result, the network's bandwidth decreases due to the utilization of only five channels to fulfill all service requirements.