(1+\underline{\Gamma})^{c_l^m} \nonumber \\
 &\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 (M+1) \sum_{l=1}^L \underline{G}^l (1+\underline{\Gamma})^{\frac{\sum_{m=0}^Mc_l^m}{M+1}}\nonumber \\
 &\hspace{-0.1in} \ge& \hspace{-0.1in} N_0  (M+1) \sum_{l=1}^L \underline{G}^l(1+\underline{\Gamma})^{\frac{l-1}{M+1}}. \label{eq:LBound}\end{aligned}$$

In (\[eq:LBound\]), the first inequality is from the definition of $\underline{G}^l$. The second inequality is due to the definition of $\underline{\Gamma}$. The third inequality is due to the fact that $(1+\Gamma)^{c_l^m}$ is a convex function. The fourth inequality is because that each level must be transmitted by at least one BS. Thus for each level $l$, there is at least one $c_l^m=c_{l-1}^m+1$ for some $m$. It follows that the sum $\sum_{m=0}^M c_l^m$ should be greater than $l-1$. Therefore, (\[eq:LBound\]) provides a lower bound for (\[eq:ObjFun2\]).

Furthermore, by defining $\underline{G} = \min_{l\in\{1,\cdots,L\}} \left\{ \underline{G}^l \right\}$, we can obtain a looser lower bound from (\[eq:LBound\]) as $$\sum_{m=0}^M Q_1^m \geq N_0 \underline{G} (M+1) \frac{(1+\underline{\Gamma})^{\frac{L}{M+1}}-1}{(1+\underline{\Gamma})^{\frac{1}{M+1}}-1}.$$

### A Simple Heuristic Scheme \[subsec:heuristic\]

We first describe a greedy heuristic algorithm that solves OPT-Power with suboptimal solutions. With this heuristic, each user compares the channel gains from the MBS and the FBS’s. It chooses the BS with the best channel condition to connect to, thus the values of the binary variables $I_m^k$ are determined. Once the binary variables are fixed, all the subsets $\mathcal{U}_l^m$’s are determined. Starting 