## 4 Proposed Scheme
### Description of the Steps of Allocating Charging Task  
The algorithm for allocating charging tasks will be introduced in the following subsections. Firstly, the initial cover sets are constructed. The proposed scheme constructs some matching pairs sets by the following algorithm. Then, allocating which nodes to drones or to the vehicle needs to be discussed. In this step, the matching pair with the longest distance is selected first, and the two paired nodes are removed from the set. Iterate the above process until the number of selected pairs achieves the number of drones being carried by the vehicle, or the node set becomes empty or a single node. In each matching pair, the node closer to the base station is allocated to the vehicle, and the other node is allocated to a drone. Lastly, a final trajectory for the vehicle is established. After allocating the nodes, the schedule of the assigned nodes for the vehicle is planned with NJNP starting from the base station, which always considers the nearest sensor to be charged first. (1) Construct the initial cover sets. This involves creating several initial cover sets based on the nodes with charging requests. To simplify the scheduling problem, we have defined a formal relationship called a \"matched pair\", for releasing drones efficiently. Definition 1 (matched pair): If two nodes \(s_{u}\) and \(s_{v}\), have charging requests, such that \(dist(s_{u},s_{v})\leq D_{\text{{dmax}}}\), then these nodes form a matched pair, denoted as (\(s_{u},s_{v}\)). Figure 3 provides an example of a matched pair of nodes. Specifically, node \(v_{i}\) and \(u_{j}\) form a matched pair in Figure 3a, but they do not form a matched pair in Figure 3b. Additionally, Figure 4 demonstrates several matched pairs, including (\(u_{1},v_{1}\)), (\(u_{1},v_{2}\)), (\(u_{2},v_{2}\)), (\(u_{3},v_{3}\)), (\(u_{4},v_{4}\)), (\(v_{1},v_{5}\)). It is evident that the scheduling problem in this step can be reduced to a set cover-like problem. Definition 2 (Cover): If nodes \(v_{i}\) and \(u_{j}\) formed a matched pair, we said that they cover each other. Figure 2: A near-optimal route for the vehicle is established by assigning \(s_{i}\) to a drone, which enables the vehicle to traverse along the tour path \(s_{\text{o}(0)}\)\(\rightarrow\)\(s_{\text{o}(1)}\)\(\rightarrow\)\(s_{\text{o}(2)}\)\(\rightarrow\)\(\ldots\)\(\rightarrow\)\(s_{\text{o}(m)}\)\(\rightarrow\)\(s_{\text{o}(m+1)}\)\(\rightarrow\)\(\ldots\)\(\rightarrow\)\(s_{\text{o}(n)}\). **Definition 3** (Set cover-like problem). : _Given a node set \(S\), the set cover-like problem requires finding a subset \(C\) of set \(S\) to satisfy the following equation:_  
\[\underset{u\in C}{\cup}A_{u}=S-C \tag{13}\]  
_where \(A_{u}\) represents the set of nodes covered by node \(u\)._  
Algorithm 2 describes the process for creating the cover sets. The computational complexity of this algorithm is \(O(n^{2})\), where \(n\) is the size of sensor nodes in the deployed network. (2) Assigning the nodes to drones or to the vehicle  
This step adopts a greedy algorithm called _long-distance first matching_ (LDFM) to select \(m\) matching pairs after computing the Euler distance of each pair. For example, as shown in Figure 4, the matched pairs from the cover set are \(\{(u_{1},v_{1})\), \((u_{1},v_{2})\), \((u_{2},v_{2})\), \((u_{3},v_{3})\), \((u_{4},v_{4})\), \((v_{1},v_{5})\)]. After computing the Euler distance of each pair, five matching pairs should be selected at most. Figure 4: The nodes with charging requests can be initially matched to form six matching pairs. These matched pairs include \((u_{1},v_{1})\), \((u_{1},v_{2})\), \((u_{2},v_{2})\), \((u_{3},v_{3})\), \((u_{4},v_{4})\) and \((v_{1},v_{5})\). Figure 3: There are two possible cases regarding matched pairs. (**a**) Nodes \(v_{i}\) and \(u_{j}\) form a matched pair, and (**b**) nodes \(v_{i}\) and \(u_{j}\) do not form a match pair. The pair set \(R\) is equal to \(\underset{u\in C,v\in A_{u}}{\cup}(u,v)\).