## 3 A Feasible Solution and the Mathematical Formulations
### Mathematical Formulations  
The fractional mixed-integer nonlinear model can, thus, be written as follows  
\[F_{1}:\min_{\{x,y\}} \frac{\sum_{i,j\in V}D_{ij}x_{ij}}{\sum_{ \begin{subarray}{c}(i\
eq j )\\ (i\
eq j)\end{subarray} }D_{ij}x_{ij}+\sum_{ \begin{subarray}{c}i,j\in V\\ (i<j)\end{subarray} }D_{ij}y_{i}y_{j}} \tag{1}\] \[\text{st}: \sum_{j\in V}x_{ij}+y_{i}=1,\quad\forall i\in V,(i\
eq j)\] (2) \[\sum_{j\in V}y_{j}=K\] (3) \[x_{ij}<=y_{j},\quad\forall i,j\in V,(i\
eq j)\] (4) \[x_{ij}<=1-y_{i}y_{j},\quad\forall i,j\in V,(i\
eq j)\] (5) \[x\in\{0,1\}^{|V|^{2}},y\in\{0,1\}^{|V|} \tag{6}\]  
It should be noted that we are referring to a fractional programming model (1) that is very hard to deal with. As such, we cannot solve it directly, but we can transform it to derive two quadratic models. The aim is to minimize the latency between the connections between switches and controllers concerning the latency of all connections in the network. In constraint (2), it is allowed that the connection of a node, \(i\in V\), can only be assigned to a single node, \(j\in V\), provided that \(i\
eq j\), with the only exception being when that node is a controller without connections. Constraint (3) ensures that the number of controllers equals \(K\). Constraints (4) impose that the connections can only occur if the destination node of that link is a controller. In constraint (5), it is ensured that two controllers cannot communicate directly. Finally, constraint (6) denotes the domains of the decision variables. Now, the transformation of the fractional model (1)-(6) into the intermediate model (7)-(9) can be derived by imposing an upper bound variable, \(t\), in the constraint (8). Furthermore, the domain definition of the variable, \(t\), is added in constraint (9). \[\min_{\{x,y,t\}} t \tag{7}\] \[\text{st}: \sum_{ \begin{subarray}{c}i,j\in V\\ (i\
eq j)\end{subarray} }D_{ij}x_{ij}\] (8) \[\sum_{ \begin{subarray}{c}i,j\in V\\ (i\
eq j)\end{subarray} }D_{ij}x_{ij}+\sum_{ \begin{subarray}{c}i,j\in V\\ (i<j)\end{subarray} }D_{ij}y_{i}y_{j}\leq t\] \[\sum_{j\in V}x_{ij}+y_{i}=1,\quad\forall i\in V,(i\
eq j)\] \[\sum_{j\in V}y_{j}=K\] \[x_{ij}<=y_{j},\quad\forall i,j\in V,(i\
eq j)\] \[x\in\{0,1\}^{|V|^{2}},y\in\{0,1\}^{|V|},t\in[0,\infty) \tag{9}\]  
The first equivalent model that we consider and evaluate by using the benchmark and random instances is presented below. Compared to the previous intermediate model, constraint (8) is replaced by constraints in (11) and (12). In addition to removing the fractional part of (8), the quadratic term \(y_{i}y_{j}\) is also replaced using variable \(z_{ij}\) for all \(i,j\in V,(i<j)\). This is carried out to avoid having a cubic term in the third sum of constraint (11). Finally, the domain of the decision variable \(z\) is included in constraint (13). \[M_{1}:\min_{\{x,y,t,z\}} t \tag{10}\] \[\text{st}: \sum_{ \begin{subarray}{c}i,j\in V\\ (i\
eq j)\end{subarray} }D_{ij}x_{ij}-\sum_{ \begin{subarray}{c}i,j\in V\\ (i\
eq j)\end{subarray} }D_{ij}tx_{ij}-\sum_{ \begin{subarray}{c}i,j\in V\\ (i<j)\end{subarray} }D_{ij}tz_{ij}\leq 0\] (11) \[\sum_{j\in V}x_{ij}+y_{i}=1,\quad\forall i\in V,(i\
eq j)\] \[\sum_{j\in V}y_{j}=K\] \[x_{ij}<=y_{j},\quad\forall i,j\in V,(i\
eq j)\] \[z_{ij}=y_{i}y_{j},\quad\forall i,j\in V,(i<j)\] (12) \[x_{ij}<=1-y_{i}y_{j},\quad\forall i,j\in V,(i\
eq j)\] \[x\in\{0,1\}^{|V|^{2}},y\in\{0,1\}^{|V|},t\in[0,\infty),z\in\{0,1 \}^{\frac{|V|(|V|-1)}{2}} \tag{13}\]  
Another variant of the fractional model is also presented below, where constraint (11) is replaced by the constraints (15)-(17), and the domain definitions of variables \(z_{1},z_{2}\) are added in constraint (22). \[M_{2}:\min_{\{x,y,t,z_{1},z_{2}\}} t \tag{14}\] \[\text{st}: z_{1}=\sum_{ \begin{subarray}{c}i,j\in V\\ (i\
eq j)\end{subarray} }D_{ij}x_{ij}\] (15) \[z_{2}=\sum_{ \begin{subarray}{c}i,j\in V\\ (i\
eq j)\end{subarray} }D_{ij}x_{ij}+\sum_{ \begin{subarray}{c}i,j\in V\\ (i<j)\end{subarray}