## 2 System Model and Problem Formulation
### Problem Formulation  
In this paper, we assume that any group of BSs use the maximum ratio transmission (MRT) [39] to transmit signals to their served user. Then, we can write the precoded weight as  
\[w_{m,n}=\sqrt{P_{m,n}}\frac{H_{m,n}}{\sqrt{\sum_{m\in\mathcal{M}} \gamma_{m,n}|H_{m,n}|^{2}}}, \tag{8}\]  
where \(P_{m,n}\) denotes the transmit power of the BS \(m\) to user \(n\). Substituting (3), (5) and (8) to (6), we can obtain the average SINR received by user \(n\) as  
\[\lambda_{n}=\frac{\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}P_{m,n}d_{m,n}}{ \sigma^{2}+\sum\limits_{j\in\mathcal{N},j\
eq n}\frac{\sum\limits_{m\in \mathcal{M}}\gamma_{m,j}P_{m,n}d_{m,j}}{\sum\limits_{m\in\mathcal{M}}\gamma_{m,j}d_{m,j}}}, \tag{9}\]  
where \(d_{m,n}=\alpha_{m,n}^{2}+L\bigg{(}\frac{\pi\sqrt{\pi}}{4}\alpha_{m,n}q_{m,n,n} +\sum\limits_{r\in\mathcal{N},r\
eq n}q_{m,r,n}^{2}\bigg{)}+\frac{L^{2}\pi^{2 }}{16}q_{m,n,n}^{2}\), \(u_{m,n}=\alpha_{m,n}^{2}+L\sum\limits_{r\in\mathcal{N}}q_{m,r,n}^{2}\) and \(d_{m,j}=\alpha_{m,j}^{2}+L\bigg{(}\frac{\pi\sqrt{\pi}}{4}\alpha_{m,j}q_{m,j,j} +\sum\limits_{r\in\mathcal{N},r\
eq j}q_{m,r,j}^{2}\bigg{)}+\frac{L^{2}\pi^{2 }}{16}q_{m,j,j}^{2}\). \(q_{m,r,n}=\beta_{m,r}\eta_{r,n}\) is the average path gain between BS \(m\) and user \(n\) via RIS \(r\). The derivation process can be referred to in Appendix A of [32], which will not be repeated here for brevity. According to the above, our objective is to optimize the transmit powers of the BSs and the BS selections to achieve the maximum weighted sum SINR for all users. Denote \(a_{n}\geq 0\) as the weighting factor to control the scheduling priority of user \(n\) and a larger value of \(a_{n}\) signifies a higher priority for information transmission to user \(n\). Based on (9), the weighted sum SINR received by users can be expressed as  
\[\sum\limits_{n\in\mathcal{N}}a_{n}\lambda_{n}=\sum\limits_{n\in \mathcal{N}}\frac{a_{n}\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}P_{m,n}d_{m,n}}{ \sigma^{2}+\sum\limits_{j\in\mathcal{N},j\
eq n}\frac{\sum\limits_{m\in \mathcal{M}}\gamma_{m,j}P_{m,j}u_{m,n}d_{m,j}}}. \tag{10}\]  
Accordingly, we can formulate the optimization problem as  
\[\mathrm{(P0)}:~{}\max\limits_{\Gamma,\mathrm{P}} \sum\limits_{n\in\mathcal{N}}a_{n}\lambda_{n}\] (11) s.t. \[\sum\limits_{n\in\mathcal{N}}\gamma_{m,n}\leq 1,~{}\forall m\in \mathcal{M}, \tag{12}\] \[\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}\geq 1,~{}\forall n\in \mathcal{N},\] (13) \[P_{m,n}\leq P_{m}^{c},~{}\forall m\in\mathcal{M},~{}\forall n\in \mathcal{N},\] (14) \[\gamma_{m,n}\in\{0,1\},~{}\forall m\in\mathcal{M},~{}\forall n\in \mathcal{N}, \tag{15}\]  
where \(P_{m}^{c}\) represents the available transmit power budget of each BS. The BS's transmit power and BS selections are denoted by \(\mathbf{P}=\{P_{m,n}\}\) and \(\mathbf{\Gamma}=\{\gamma_{m,n}\}\), respectively. The weighted sum-SINR maximization problem (P0) is a complex task that involves optimizing a nonlinear sum-of-ratios function [40], which involves maximizing a sum of several fractional functions. This non-convex optimization problem is notoriously difficult to solve optimally. Furthermore, the problem requires the joint optimization of the BS transmit power and BS selection, where the latter aspect involves binary/integer variables, i.e., \(\gamma_{m,n}\). So, we decompose it into following two subproblems to solve this problem with low complexity. * BS Transmit Power Subproblem: Given the fixed BS selections \(\mathbf{\Gamma}\), we focus on optimizing the transmit power of each BS, while ensuring it falls within the allocated budget \(P_{m}^{c}\). Then, the BS power problem can be formulated by \[(\mathrm{P1}):\ \ \underset{\mathbf{P}}{\text{max}} \sum\limits_{n\in\mathcal{N}}a_{n}\lambda_{n}\] (16) s.t.