## 1 Introduction
### Related Works  
For several years, numerous algorithms have been developed for solving the TSP, mainly the symmetric TSP. A classification of some of these algorithms is presented in Figure 3.  
The exact algorithms aim to find the optimal solution to a TSP. For example, the Brute Force algorithm calculates all the possible permutations, whereas the Cutting Plane [27], the Branch-and-Bound [28] and the Branch-and-Cut methods [29] limit the number of possibilities to reduce the calculation time. Concorde is an exact solver which combines several of these techniques, and is able to solve TSP with numerous cities (tens of thousands) [30]. The limitation of these exact algorithms is however the calculation time which  
Figure 3: Algorithms for solving TSP.  
Figure 2: Problem statement in 2D: find the optimal trajectory passing through the circular disks representing the radio communication ranges of the buried sensor nodes at the altitude h.  
can be rapidly become excessive in case of numerous cities to be visited. A strategy can be to parallelize the algorithms and distribute the calculation on several CPU [31].  
Another approach consists to develop approximate algorithms, with the objective to find a sub-optimal solution in a limited amount of time. As depicted in Figure 3, the approximate algorithms can be classified into three main groups, those based on constructive heuristics, those based on improvement heuristics and those based on meta-heuristics, see [32]. The principle of constructive heuristics is to start from an empty tour and iteratively extend the tour, for example by incorporating the nearest neighbor. The strategy of improvement heuristics is different because they start from a complete tour, which is progressively improved, for example using local permutations (e.g., K-opt [33], Lin-Kernigan-Helsgaun [34]). Meta-heuristic algorithms are generic algorithms able to deal with a broad range of problems. They have generally better performances than the heuristics algorithms, in particular by enabling to consider more complex and large TSP. They are usually based on the evolution of one or more starting populations (e.g., Genetic Algorithm [35], Ant Colony Optimization [36], Particle Swarm optimization [12]).  
In the literature, the CETSP has been mainly addressed with meta-heuristic approaches. The difficulty of that problem is that the target areas are continuous, as at each point is associated a neighborhood, leading that even exact approaches are not easily applicable. The shapes of the neighborhoods are most of the time considered as discs with the same dimensions.  
Initially, the CETSP was addressed by [37; 38] to find the shortest route for a truck to perform automatic meter reading of customers from radio communication systems (e.g., remote electricity or water consumption measurements). Clustering-based and convex hull-based algorithms were proposed to group the customer nodes before applying a TSP solver. This approach is mainly based on the search for the adequate definition and position of groups of nodes, called supernodes, i.e., circles with a predetermined radius covering several customers. A CETSP with time window (CETSP-TW) was studied by [18] to take into account that each customer is available only for a limited period of time during the day. Ref. [39] introduced the Steiner Zones (intersection of the neighbhohoods of degree k) and presented different methods to reduce the graphs. Ref. [40] proposed a branch-and-bound algorithm to address the CETSP in two and three dimensions, and [17] presented a method to estimate previously a tour length from the knowledge of several variables. Refs. [41; 42] proposed discretization schemes to define first a finite number of points on the circumferences and inside the neighborhoods (hitting points), and next apply a TSP algorithm. Ref. [43] assumed disjoint regions and applied first a TSP algorithm to next modify the hitting points.