## 2 Problem Modeling
### Abstract Modeling Based on Graph Coloring  
Graphs are one of the most commonly-used tools for modeling the frequency assignment problems, because of the relation of this problem to the graph coloring problem, which has been widely studied by the mathematical community [27].  
In graph coloring, an abstract graph is considered, defined by a set of vertices along with some edges connecting them, and the objective is to assign one color to each vertex, in such a manner that the minimum possible number of colors should be used, while avoiding monochromatic edges.  
In the commonly-used model, graph nodes represent elements that should be assigned a frequency, while edges represent element pairs that should not be assigned the same frequency. This way, colors act as frequencies, and hetero-chromatic edges guarantee element pairs with different frequencies.  
Although widely used, [PERSON] _et al._[28] conclude that the model is not accurate enough, because it does not reflect all of the information. For instance, the authors suggest that the information regarding adjacent channel interferences should be incorporated into the graph. In this work, we do so and use a graph showing greater fidelity to the problem modeled.  
We can distinguish three types of vertices and two types of edges as graph elements. Regarding vertices, we consider access points (AP), sensors or cameras (C) and other wireless devices (D). Regarding edges, on the one hand, every wireless device (C or D) will be associated with its closest  
Figure 1: Wireless surveillance sensor network (WSSN) architecture.  
AP, trying to reflect that it will use the channel assigned to its closest AP. On the other hand, we will also link some node pairs when the distance between them is below the corresponding interference radius \(R\) (to reflect interferences): AP-AP pairs will be linked provided that the distance condition is met, AP-C (AP-D) pairs only when the camera (device) is not associated with that AP and C-C (D-D) pairs only if both cameras (devices) are associated with different APs, as the communications among the elements connected to the same AP are coordinated.  
Furthermore, to model the interference power between two elements, we weigh each edge based on three factors. First, we consider a weight for each color pair \(ij\), which can be understood as the interference between color \(i\) and color \(j\). It is worth noting that the usual coloring problem only takes into account the particular case of interferences between vertices of the same color, while our extension of the problem allows considering also interferences between adjacent colors or colors in a certain distance range.  
Second, we weigh the previous weights, by introducing the distance between edges' endpoints. This way, the weight assigned to a colored edge \(ij\) will be different depending on how far apart its endpoints are. This represents another extension to the usual coloring problem, because now vertices have also certain positions, and this means that our graph is no longer abstract, but geometrical.  
Third, we include the effect of the amount of data into the weights, including a factor that accounts for the fact that a higher bandwidth data flow will occupy the wireless channel a higher fraction of the time.