# Expected output Player norms:[1.77552809 1.69189243 1.68448805] ```
### Experimental Analysis
## 6 Complexity Analysis and Scalability Issues

In this section, we discuss the complexity of the proposed model and its potential computational limitations. The model has two components: a GCN and MILP. The GCN component scales to the size of the graph, while the MILP component struggles with the exponential complexity of large problems. The computational limitations and the scaling behavior of this model can be decomposed into the following items. First, we discuss its scaling behavior in terms of the following:

* Number of Variables: The number of decision variables \(x_{ij}\) grows as \(n\) (the number of players) multiplied by \(m\) (the number of teams). * Number of Constraints: The number of constraints grows linearly with the number of players and the number of teams. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Genetic Algorithm with**} \\  & **Genetic Algorithm** & **Local Search and Adaptive** & **GCN-MILP** \\  & & **Penalties** & \\ \hline Mean Skill & 1.7840 & 2.0660 & 0.6222 \\ Distribution & 4.8018 & 6.0820 & 1.6956 \\ Variance & 0.8025 & 0.645 & 0.1052 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A comparative analysis. * Computational Complexity: The worst case to solve using the MILP in our TFP settings is the exponential one. Second, we present the computational limitations for large teams, which are as follows:
* Memory: MILP solvers need to store the problem's data, i.e., the search tree generated during the branch-and-bound process. For very large problems, this could be affected by memory limitations. * Solution or Convergence Time: Finding the optimal solution could take a long time when dealing with large problems. * Solver Performance: Some solvers are more efficient than others. Thus, choosing the right solver is crucial. If you have a large-scale problem, Gurobi [30] and CPLEX [31] are generally worth investing in, as they are both commercial. However, some scalability solutions can be used to handle extremely large problems. For example, high-performance computing resources, e.g., parallel computing or cloud computing, might be used to solve the MILP within a reasonable timeframe. In addition, if we are not trying to find the absolute optimal solution and near-optimal or good solutions are suitable, then opting for alternative approaches like heuristics and metaheuristics could be the right choice. Next, we present a detailed complexity analysis of the proposed GCN-MILP model. The time complexity for the GCN stage is \(\mathcal{O}(l\times m\times d_{max}^{2})\), where \(l\) denotes the number of layers, \(m\) denotes the number of edges, and \(d_{max}\) signifies the maximum feature dimension across all layers. This relation is computed by considering that the time complexity of each layer is expressed as \(\mathcal{O}(m\times d_{in}\times d_{out})\), where \(d_{in}\) and \(d_{out}\) denote the input and output feature dimensions for that layer, respectively. If we assume that \(d_{in}\) and \(d_{out}\) are approximately equal and correspond to \(d_{max}\), the complexity of each layer simplifies to \(\mathcal{O}(m\times d_{max}^{2})\). Then, for \(l\) layers, the total time complexity is \(\mathcal{O}(l\times m\times d_{max}^{2})\). The space complexity for GCN is \(\mathcal{O}(n\times d_{max}+m+l\times d_{max}^{2})\), where \(n\) denotes the number of nodes. This space complexity formula accounts for node feature storage (\(n.d_{max}\)) for all nodes; graph adjacency storage (\(m\)), needed to store the adjacency information of the graph, which is in a sparse format; and weight matrices for all layers (\(l\times d_{max}^{2}\)). As for MILP, MILP problems are NP-hard, as they grow exponentially with the number of variables and constraints. The worst-case complexity for an MILP problem is \(\mathcal{O}(2^{n})\), where \(n\) is the number of binary variables. ## 7 Discussion

Using GNNs to better represent relationships and preferences in team formation has the potential to produce a solution that best aligns with the needed quality.