## 3 Network Description
Each sensor \(s\in\mathcal{S}\) is the source of packets (for example, containing measurement data that the sensor collect from the environment) that are to be delivered to multiple destinations specified by the set of destinations \(\mathcal{D}(s)\subseteq\mathcal{D}\). We assume that the sensors are capable of both generating and transiting packets, while the destination nodes only terminate traffic, i.e., they are not able to transit packets; clearly, the transit nodes do not generate or terminate packets, and only transit them. The packets that are generated by sensor \(s\) are transmitted to their destinations along paths of a multicast tree \(\mathcal{B}(s)=(\mathcal{V}(s),\mathcal{A}(s))\), where \(\mathcal{V}(s)\subseteq\mathcal{V}\) is the set of its nodes while \(\mathcal{A}(s)\subseteq\mathcal{A}\) is the set of its arcs (\(|\mathcal{A}(s)|=|\mathcal{V}(s)|-1\)). Clearly, such a tree is routed at \(s\) and \(\mathcal{D}(s)\) are its leaves. Note that, at each node \(v\) of the multicast tree \(\mathcal{B}(s)\) (except for the leaves), the packets are, in general, transferred to more than one neighboring node, and this is the place when we take advantage of the broadcast nature of radio transmissions, since the packet broadcast from a node \(v\) reaches all of the nodes within its transmission range of the node \(v\) anyway. We consider TDMA-based WSNs, which means that the packets are transmitted within equally long time slots that are grouped into consecutive TDMA frames of \(T\) time slots each. We assume that each sensor sends out the measurement data at the beginning of each consecutive frame. The frames are repeated periodically and, hence, the transmission patterns that are applied in the consecutive frames are identical, i.e., the set of broadcasting nodes as well as the set of receiving nodes in \(t\)th slot (\(t=1,2,\ldots,T\)) of each frame are the same. A pair composed of the set of (simultaneously) broadcasting nodes and the sets of simultaneously receiving nodes defined for each broadcasting node will be called compatible set (c-set in short) and they will be denoted by \(c\). In the following, \(\widetilde{\mathcal{C}}\) will denote the family of all c-sets for a given network (note that the size of this family grows exponentially with the network size), and \(\mathcal{C}\) a selected subfamily of \(\widetilde{\mathcal{C}}\). The set of nodes broadcasting in c-set \(c\) will be denoted by \(\mathcal{W}(c)\), \(m(c,w)\), \(w\in\mathcal{W}(c)\) denotes MCSs assignment, and the set of nodes receiving from node \(w\in\mathcal{W}(c)\) in c-set \(c\) will be denoted by \(\mathcal{U}(c,w)\). Note that any c-set \(c\) is valid if, and only if, each of the nodes in \(\mathcal{U}(c,w)\) can decode the signal from \(w\), i.e., the nodes receiving from \(w\) are not interfered by the signals that are simultaneously broadcast from the nodes in \(\mathcal{W}\setminus\{w\}\). Such a condition can be expressed in the following formula:  
\[\frac{p(w,u)}{\eta+\sum_{v\in\mathcal{W}\setminus\{w\}}p(v,u)}\geq\gamma(m(c,w )),\ w\in\mathcal{W},u\in\mathcal{U}(c,w). \tag{1}\]  
The above c-set description can by formally expressed by the set of linear inequalities imposed on the variables that characterize the sets \(\mathcal{W}(c)\) and \(\mathcal{U}(c,w)\), MCS assignment \(m(c,w)\), \(w\in\mathcal{W}(c)\), and the transmission power of each node; such a formal definition will be introduced in the following part of the article.