## 3 The Proposed Scheme for WSP Minimization
### Link Matching for Multiple PLs and SLs  
Based on the above derivation, the spectrum allocation and power control scheme via [PERSON]'s method for one matched PL-SL pair is taking shape. It should be noted that this scheme is centralized and requires a network central controller. In this subsection, we discuss the link matching to extend this scheme to general scenarios with multiple PLs and SLs.  
In our system--shown in Figure 1--the public PR, i.e., the BS, performs as a central controller and determines the matching between \(\mathcal{P}\) and \(\mathcal{S}\) through the KM algorithm [36]. We assume complete CSI is available at the BS. According to the matching rule C5, we can formulate the link matching, which aims to minimize the system WSP as a minimum cost bipartite matching with the weights \(\{w_{mn}\}\) obtained from Equation (7). Through the proposed spectrum allocation and power control strategy described in the previous subsection, a weight matrix can be formed at the PR as follows:  
\[\mathbf{W}=\begin{bmatrix}w_{11}&w_{12}&\cdots&w_{1N}\\ w_{21}&w_{22}&\cdots&w_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ w_{M1}&w_{M2}&\cdots&w_{MN}\end{bmatrix}. \tag{24}\]  
To allocate wireless resources for \(M\) PLs and \(N\) SLs, the central controller, namely, the BS, should collect the instantaneous CSI via control channels at the beginning of every frame. Suppose a CSI report contains \(A\) bits. The CSI of the SLs is required but the CSI of the PLs is not; therefore, the overhead for the CSI reporting is \(MNA\) bits. As \(M\) or \(N\) increases, the CSI overhead linearly increases. One of our future research directions will be how to reduce CSI feedback and implement our scheme in partial CSI situations.  
The time complexity of the KM algorithm is \(O(\max\{M,N\}^{3})\). In typical 5G eMBB scenarios, the user capacity of a single cell usually ranges from hundreds to thousands. As the number of links increases, the execution time of the KM algorithm will also increase rapidly. Thus, the scalability of the proposed method in large-scale network scenarios is an important topic that concerns its practicality. In our prior work [37], we introduced cooperative link sets to accelerate the proposed deep reinforcement learning (DRL)-based hybrid centralized-distributed scheme and reduce signaling exchange. Unfortunately, similar mechanisms are challenging to apply to the proposed scheme in this paper.  
Distance-constrained spanning trees (DCSTs) are vital for optimizing spectrum allocation and power control. They ensure efficient connectivity while minimizing interference and power consumption. In overlay spectrum sharing, structuring users within specific distance constraints enhances signal quality, upholds the QoS requirements, and reduces unnecessary power expenditure. This approach aligns with our goal of minimizing WSP. In [38], the authors address fault tolerance and robust communication networks by constructing multiple disjoint spanning trees that adhere to strict distance constraints.  
On the other hand, DCSTs help reduce computational complexity by limiting the number of potential connections. This limitation accelerates the KM algorithm for link matching and enhances the convergence speed of [PERSON]'s method. To address scalability challenges in ultra-dense networks, meta-heuristic approaches (e.g., genetic algorithms, particle swarm optimization, or simulated annealing) can efficiently find near-optimal solutions. This integration ensures fair energy distribution and improved spectrum reuse without excessive computational overhead. By structuring the network into clusters or subgraphs, DCSTs combined with meta-heuristic methods provide a more scalable and computationally feasible approach to real-time spectrum allocation and power control in next-generation wireless networks. In our future research, we will aim to further address the scalability issues of the proposed scheme through the use of DCSTs.