*2.4. Task Identification Method*

The task identification problem is addressed following a frontier point approach [5] where the *free* cells (cf. Section 2.1) that belong to a frontier are over labelled as *frontier points (FP)*. Besides, the resulting set of *FP* cells is clustered (using procedures such as *K-Means* [41] or *Affinity Propagation* [42]) in order to identify the cells that better represent each frontier, defining a set of tasks (in the remainder of the document, the terms *task* and *target* are used indistinctly) *<sup>T</sup>* <sup>=</sup> {*T*1, *<sup>T</sup>*2, ... , *TN*} | *Tj* <sup>∈</sup> <sup>R</sup>2, <sup>∀</sup>*<sup>j</sup>* <sup>∈</sup> {1 ... *N*}. Thus, *T* represents, at each moment, the smallest set of promising locations that the robots could be interested in visiting to explore all frontiers. In Figure 3 these *task* cells are coloured in yellow.

**Figure 3.** Frontier points. The different cell types are identified according to the following colour code: dark blue cells are **Obs**tacles, light blue cells are **Unk**nown, green cells are **Free**, orange cells are **FP** cells, and yellow cells are *tasks*.

### *2.5. Multi-Robot Task Allocation Problem—MRTA*

Following the classification proposed in [10], the *MRTA* problem to be tackled is described as a *single-task robots (ST)*, *single-robot tasks (SR)*, and *instantaneous assignment (IA)* problem. *ST* means that each *robot* is able to visit at most one *task* at a time. *SR* means that each *task* requires only one *robot* to be explored. *IA* means that the available information about the robots, the tasks, and the environment permits only an instantaneous allocation of tasks to robots, preventing the possibility to plan future allocations. Additionally, an *ST-SR-IA* can be formulated as an instance of the well known *Optimal Assignment Problem (OAP)* as follows. Given *M* robots, *N* tasks, and utility estimates *U* for each *MN* possible robot-task pair, the goal is to assign tasks to robots so as to maximise overall expected utility. Finally, from an *Integer Linear Programming* perspective, the problem can be formalised as: Find the *MN* non-negative integers *αij* that maximise (2).

$$\sum\_{i=1}^{M} \sum\_{j=1}^{N} \alpha\_{ij} \, \text{l}I\_{ij} \tag{2}$$

s.t.

$$\sum\_{i=1}^{M} \mathfrak{a}\_{i\bar{j}} = 1, 1 \le j \le N$$

$$\sum\_{j=1}^{N} \mathfrak{a}\_{i\bar{j}} = 1, 1 \le i \le M$$
