## 3 Network Description  
In this section, we present the basic assumptions of our optimization model together with the notations that are summarized in Table 1.  
The considered WSN is modeled by means of a directed graph \(\mathcal{G}=(\mathcal{V},\mathcal{A})\), where \(\mathcal{V}\) is the set of nodes and \(\mathcal{A}\subseteq\mathcal{V}\times\mathcal{V}\setminus\left\{(v,v):v\in \mathcal{V}\right\}\) is the set of directed links (arcs). The beginning of arc \(a\) will be denoted by \(b(a)\), while its end by \(e(a)\). We assume that \((w,v)\in\mathcal{A}\) if, and only if, node \(v\) is in the transmission range of node \(w\), i.e., when the signal-to-noise-ratio (SNR) condition is satisfied at node \(v\) when node \(w\) is transmitting and no other node is transmitting at the same time. Such a condition can be formally expressed by the following formula: \(\frac{p(w,v)}{q}\geq\gamma(m)\), where \(p(w,v)\) is the power of signal transmitted by node \(w\) and received by node \(v\), \(\eta\) is the noise power, \(m\in\mathcal{M}\) is a modulation and coding scheme (MCS) used by node \(w\) (\(\mathcal{M}=\{1,2,\ldots,M\}\) denotes the set of available \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(\mathcal{V}\) & set of nodes \\ \(\mathcal{S},\mathcal{D},\mathcal{Q}\) & sensors, destinations, transit nodes, respectively \\ \(\mathcal{A}\) & set of directed radio links (arcs); \(a=(b(a),e(a))\) \\ \(\delta^{+}(v)\) & set of nodes connected with \(v\) by its outgoing arcs \\ \(\delta^{-}(v)\) & set of nodes connected with \(v\) by its incoming arcs \\ \(p(v,w)\) & power received at node \(w\) from transmitting node \(v\) \\ \(\eta\) & noise power \\ \(m\) & modulation and coding scheme \\ \(\mathcal{M}\) & set of MCSs (\(\mathcal{M}:=\{1,2,\ldots,M\}\)) \\ \(\gamma(m)\) & SINR threshold for MCS \(m\) \\ \(B(m)\) & bitrate in (Mbps) for MCS \(m\) \\ \(c\) & compatible set (\(c\)-set in short) \\ \(m(c,w)\) & number of the MCS assigned to node \(w\) in c-set \(c\) \\ \(\tilde{\mathcal{C}}\) & family of all c-sets \\ \(\mathcal{W}(c)\) & set of nodes transmitting when c-set \(c\) is used \\ \(\mathcal{U}(c,w)\) & set of nodes that receive signal from \(w\in\mathcal{W}(c)\) \\ \(\mathcal{C}\) & selected subfamily of \(\tilde{\mathcal{C}}\) \\ \(\mathcal{C}(a)\) & subset of c-sets in C broadcasting over arc a \(\mathcal{C}(a)=\mathcal{C}(b(a),e(a))\) \\ \(\mathcal{D}(s)\) & set of destination nodes of sensor \(s\) (\(\mathcal{D}(s)\subseteq\mathcal{D}\)) \\ \(\mathcal{B}(s)\) & multicast routing tree rooted at \(s\) with leaves \(\mathcal{D}(s)\) \\ \(\mathcal{V}(s),\mathcal{A}(s)\) & set of nodes and arcs, respectively, of \(\mathcal{B}(s)\) \\ \(\underline{B},\mathbb{Z},\mathbb{R}_{+}\) & \(\mathbb{B}=\{0,1\}\), \(\mathbb{Z}\) integers, \(\mathbb{R}_{+}\) non-negative real numbers \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation: network setting. MCSs), and \(\gamma(m)>1\) is the assumed power ratio threshold for a given MCS \(m\). Besides the power ratio threshold \(\gamma(m)\), each MCS \(m\) is also characterized by the transmission rate \(B(m)\) that is expressed in Mbps. Note that, because MCSs have different power ratio threshold, the transmission range of a given node depends on a MCS used by this node. For a more detailed description of the radio parameters, the reader is referred to Section 7.1. The set of neighboring nodes connected to \(v\) by the arcs outgoing from \(v\) is denoted by \(\delta^{+}(v):=\{w\in\mathcal{V},(v,w)\in\mathcal{A}\}\). Similarly, the set of neighboring nodes connected by the arcs incoming to \(v\) is denoted by \(\delta^{-}(v):=\{w\in\mathcal{V},(w,v)\in\mathcal{A}\}\). We assume that the network nodes are divided into three pairwise disjoint subsets: sensors \(\mathcal{S}\), destinations \(\mathcal{D}\), and purely transit nodes \(\mathcal{Q}\).