its signal can be decoded by $R_r$, i.e., $$\begin{aligned}
\label{eq:P_R_t_max}
 \frac{g_{R_t R_r} P_{R_t}^{\max}}{g_{S R_r}P_{S} +  n_{R_r}} <  \gamma_0 \; .\end{aligned}$$ but it should definitely be high enough to be decoded at the destination, i.e., $$\begin{aligned}
\frac{\gamma_0 n_D}{g_{R_t D}}\leq P_{R_t} &\leq  P_{R_t}^{\max} \label{subeq:ineq11}.\end{aligned}$$ Then, $P_S$ should be high enough, so that the transmitting signal can be successfully decoded at $R_r$. Hence, $$\begin{aligned}
\gamma_0 \left(\frac{g_{R_t R_r}}{g_{S R_r}} P_{R_t}  +  \frac{n_{R_r}}{g_{S R_r}}\right) \leq P_{S} &\leq P_{S}^{\max} \label{subeq:ineq21}\end{aligned}$$ Substituting into we have $$\begin{aligned}
 \label{eq:condition}
\gamma_0 \left(\frac{g_{R_t R_r}}{g_{S R_r}} \frac{\gamma_0 n_D}{g_{R_t D}} +  \frac{n_{R_r}}{g_{S R_r}}\right) \leq P_{S} &\leq P_{S}^{\max} \; .\end{aligned}$$ Substituting into we obtain that $$\begin{aligned}
P_{R_t}^{\max} < \gamma_0 \left[ \gamma_0 \left(\frac{\gamma_0 n_D}{g_{R_t D}}   + \frac{n_{R_r}}{g_{R_t R_r}}   \right) + \frac{n_{R_r}}{g_{R_t R_r}}   \right] \; .\end{aligned}$$

Making use of Propositions \[prop:1\] and \[prop:2\], we will now describe the $\minpow$ relay selection algorithm. The algorithm is implemented in the following steps:\
1) First, for each possible pair of relays, we carry an *IC feasibility check*, i.e., we check through if interference cancellation is feasible.\
2) If IC is *feasible*, then

-   assuming IC took place, $P_{R_t}^*$ and $P_{S}^*$ are as given in and , respectively. Note that since the feasibility criterion is satisfied, then $P_{R_t}^* \leq P_{R_t}^{\max}$. 