*5.2. Task Selection Algorithm*

The task selection process employs the multi-objective utility function Φ defined in (7) with *α*HO values dynamically adapted by (15) to solve the *MRTA* problem stated in Secion 2.5. The corresponding algorithm is sketched in Algorithm 2.



Firstly, the input parameter *R*∗ specifically corresponds to the locations of the teammates currently connected with the robot *Ri*. Next, in lines 2 and 3, both the *task* and *robot* location sets are split up into two subsets each one (assigned and unassigned items, respectively). Line 4 is in charge of taking only the unassigned tasks that are within the *HO-Threshold* scope from every robot. Afterwards, from lines 5 until 9, the path utility matrix is computed regarding all possible task-robot combinations. Next, lines 10 and 11 aim to compute the *α*HO and *β* values according to (15). The set of tasks-to-robots distributions is calculated from line 12 to 14. Finally, from line 15 to 17 all possible assignments are evaluated using the Φ function while the task corresponding to robot *i* of the best assignment is selected in line 18.

Some considerations on Algorithm 2 are hereafter discussed. Concerning the computation of the set of tasks-to-robots distributions (lines 12 to 14), it provides a way to potentially avoid falling in local minima or even taking wrong decisions. Note that the *connectivity utility* function is subject to locality conditions and thus, it is not possible to compute optimal distributions from the application of iterative polynomial-time assignation algorithms such as the *Hungarian method* [14].

On the contrary, Algorithm 2 can choose the optimum tasks-to-robots distribution by evaluating all possible *T*HO-to-*R<sup>u</sup>* distributions. Nevertheless, this process may be potentially very hard since |*ArN*HO *<sup>M</sup><sup>u</sup>* <sup>|</sup> <sup>=</sup> *<sup>N</sup>*HO! (*N*HO−*Mu*)! <sup>=</sup> <sup>Π</sup>*<sup>n</sup> <sup>m</sup>*=1(*N*HO <sup>−</sup> *<sup>m</sup>* <sup>+</sup> <sup>1</sup>) = <sup>Π</sup>*n*−<sup>1</sup> *<sup>m</sup>*=0(*N*HO <sup>−</sup> *<sup>m</sup>*) <sup>→</sup> *<sup>O</sup>*(*N*∗*M<sup>u</sup>* ). Therefore, the smaller <sup>|</sup>*T*HO<sup>|</sup> and <sup>|</sup>*Ru*<sup>|</sup> the faster the algorithm will run. In the first case <sup>|</sup>*T*HO<sup>|</sup> is bounded by pruning <sup>|</sup>*Tu*<sup>|</sup> with the help of *HO-Threshold*.

On the contrary, even being naturally bounded (|*R*|≥|*R*∗|≥|*Ru*|), the set *<sup>R</sup>* could imply a large *<sup>R</sup>u*. Besides, all efforts are to keep the fleet connected as much as possible, leading to <sup>|</sup>*R*∗|→|*R*|. Fortunately, in a fully asynchronous multi-robot system the probability of two or more robots being simultaneously making a decision is negligible.

Finally, note that Algorithm <sup>2</sup> assumes <sup>|</sup>*T*HO<sup>|</sup> <sup>&</sup>gt; <sup>|</sup>*Ru*|; otherwise the tasks-to-robots distribution cannot be computed. In such a case, the input parameters are managed in order to conduct a robots-to-tasks distribution instead. In turn, <sup>|</sup>*ArM<sup>u</sup> <sup>N</sup>*HO | does not represent a significant effort since *<sup>M</sup><sup>u</sup>* <sup>≥</sup> *<sup>N</sup>*HO holds for small values.

## **6. Baseline Statement and AAMO Approach Results**

The aims of this section are: (i) To establish a baseline on the main figure of merits that will be defined to asses the benefits of different approaches; (ii) To assess and analyse the performance of different instances of the *Auto-Adaptive Multi-Objective (AAMO)* approach (different instances—from now on—refer to different *HO-Threshold* setup values) under non-ideal communication conditions; (iii) To compare *AAMO* instances against other approaches under non-ideal communication conditions.

Regarding the first purpose, the baseline is established regarding two state-of-art approaches so that the simulation runnings concern the comparison between a *Yamauchi*-based algorithm [5] and the *minPos* algorithm [17] under ideal communication conditions. These algorithms were chosen since they are decentralised, as are the author's proposal; while *Yamauchi* is a reference on exploration and typically serves itself as a comparison baseline, the *minPos* proposal has demonstrated very good performance, outperforming other important reference algorithms.

On the contrary, regarding the *AAMO* assessment and the comparison with other approaches, the simulation runnings concern exploration missions subject to non-ideal communication conditions. In this case, the primary purpose is to understand how compromised could be the exploration time performance when the connectivity level is prioritised and to reveal possible improvements concerning previous techniques. In consequence, there are experiments which compare only the performance achieved by different instances of *AAMO*, while in other experiments, where relevant, comparison with state-of-art performance is taken into account too.
