## 4 Adaptive \(\alpha\)-Value Computation
### Task Selection Framework  
This process is always made iteratively from a list, comparing the currently best task against the rest, one by one. Therefore, without loss of generality, the most relevant aspects can be studied just analysing all the possible relations between an arbitrary pair of tasks. Regarding the distance to a specific robot location and the connectivity level (number of connections with the rest of the fleet), any task can be classified according to Table 1.  
Therefore, the meaning of these categories is straightforward: regarding the assignment of the fleet, \(Co\) means that the task location would offer to the robot at least the minimum level of connectivity (i.e., one connection to another fleet member); \(NC\) means the opposite; regarding the spatial distribution of tasks, \(Cl\) means that the task under consideration is closest to the robot than any other; \(F\) means that the task is furthest to the robot than any other task. Moreover, let \(R_{i}\) a robot and \(T_{j}\) and \(T_{k}\) two tasks such that \(class(T)\) can belong to any class defined in Table 1. In any scenario, these tasks can be related to each other according to Table 2. Given that \(T_{j}\) and \(T_{k}\) are arbitrary tasks, the matrix can be considered symmetric. Thus, taking one of the triangular matrices is enough to study all possible cases. From the lower triangular, it is possible to identify some cases where one task is better (regarding both path utility and connectivity utility) than the other. Such an example is the [Cl/Co;F/NC] where \(T_{j}\) is closer to the robot than \(T_{k}\), and it is the only one that keeps the robot connected as well. Similarly, in the [Cl/NC;F/NC] case neither task can keep the robot connected, and in consequence, the closest task \(T_{j}\) results more convenient than \(T_{k}\). Thus, in both previous cases, the criterion to choose a task is \begin{table}
\begin{tabular}{l c c} \hline \hline ConnectivityDistance & Closest (Cl) & Furthest (F) \\ \hline Connected (Co) & Cl/Co & F/Co \\ Non-Connected (NC) & Cl/NC & F/NC \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Task classification. clear: the closest task should be selected. However, in the other cases, it is not clear at all which task should be selected. In one case, [Cl/NC;F/Co], whichever selection implies either traversing longer distances or losing connectivity. In the other case, [Cl/Co;F/Co], selecting the closest task \(T_{j}\) ensures traversing the shortest path but could imply losing connectivity. By contrast, selecting the furthest task \(T_{k}\) would be acceptable only when the gain in connectivity oppose a more significant travelling effort. The human operator threshold HO-Threshold expresses the human operator criterion through a distance that represents the extra effort made by robots that the human operator is willing to accept in order to maintain or enlarge the size of the robot communication network. In other words, the human operator criterion is determined by setting the distance threshold until which the targets that preserve or enlarge connectivity are preferred over the rest, for all robots. For instance, in the [Cl/NC;F/Co] case the selection will be conditioned as follows: \(T_{k}\) will be selected if and only if the length of the shortest path between \(T_{k}\) and the robot location is less than or equal to \(HO\)-_Threshold_. \(T_{j}\) will be selected otherwise. In order to make the influence of \(HO\)-_Threshold_ clearer, an example scene is depicted in Figure 7. Note that all tasks are within the \(HO\)-_Threshold_, but only \(T_{3}\) can enlarge the connectivity level of the robot \(R_{1}\). Thus, applying Definition 1 leads to the selection of task \(T_{3}\) because it enables the robot \(R_{1}\) to travel more distance to gain connectivity. On the contrary, whether the \(HO\)-_Threshold_\(\leq\) 3, \(T_{3}\) would be no longer preferred over the rest, and consequently the closest task \(T_{2}\) would be selected instead.