## 3 1-Principal-1-Assistant Scenario
### Problem Formalization  
Let \(V=\{v_{i}\ |\ 0\leq i\leq n\}\) be the set of vertices, where \(v_{0}\) is the location of the initial vertex, and the others denote the locations of tasks, each of which contains only one task. Each vertex can be visited several times, but the task at the vertex can be served at most once. The task at vertex \(v_{i}\in V\backslash\{v_{0}\}\) has an independent deadline \(d_{i}\), which means that the task at \(v_{i}\) needs to be served before \(d_{i}\). Additionally, the initial vertex at \(v_{0}\) is associated with a deadline \(d_{0}\), which means that the principal-assistant team must return to the initial vertex before \(d_{0}\). Let \(D=\{d_{i}\ |\ v_{i}\in V,d_{i}\leq d_{0}\}\) be the set of deadlines. In such a principal-assistant collaboration system, the principal agent equipped with an assistant agent starts from the initial vertex and returns to it before \(d_{0}\), releasing and retrieving the assistant agent to serve other tasks repeatedly. The principal agent works in a connected undirected graph \(G_{1}=(V,M_{1})\), where \(M_{1}\) denotes the set of the edges between the vertices in \(V\). Let \(m_{1}(v_{i},v_{j})\) represent the minimum time cost between \(v_{i}\in V\) and \(v_{j}\in V\) in \(G_{1}\). Without loss of generality, it is assumed that \(d_{i}\geq m_{1}(v_{0},v_{i})\), and the assistant agent is often assumed to move freely from a vertex to another and spend less time than the principal agent [4]. Meanwhile, it works in a fully connected undirected graph \(G_{2}=(V,M_{2})\), where \(M_{2}\) denotes the set of edges between the vertices in \(V\). Let \(m_{2}(v_{i},v_{j})\in M_{2}\) denote the minimum time cost between \(v_{i}\in V\) and \(v_{j}\in V\) in graph \(G_{2}\) and \(m_{2}(v_{i},v_{j})\leq m_{1}(v_{i},v_{j})\). The other assumptions about the assistant agent are as follows: (1) the assistant agent has limited endurance \(\eta\); (2) the assistant agent needs to be released and retrieved at a different vertex, except when the assistant agent is released and retrieved at the initial vertex \(v_{0}\); and the assistant agent cannot reach other places except the location of the tasks being served; (3) the assistant agent needs to reach the vertex where it is retrieved before the principal agent arrives. The principal-assistant route is denoted by a route \(r=\{principal, assistant_{r}\}\), where \(principal_{r}\) is the principal agent route and \(assistant_{r}\) is the assistant agent route. Let \(N(r)\) denote the number of served vertices in the route \(r\). The principal agent route is denoted by \(principal_{r}=\{p_{i}\}\), where the principal agent departs from \(p_{0}=v_{0}\), moves from \(p_{i}\) to \(p_{i+1}\), and serves \(p_{i}\). The number of vertices in \(principal_{r}\) is denoted by \(|principal_{r}|=g\), and the number of the served vertices in \(principal_{r}\) is \(g-1\). The assistant agent route is denoted by \(assistant_{r}=\{(p_{i},p_{k},p_{j})\}\), where a tuple \((p_{i},p_{j},p_{k})\) indicates that the assistant agent is released at \(p_{i}\in principal_{r}\) (called the _released vertex_) and retrieved at \(p_{j}\in principal_{r}\) (called the _retrieved vertex_), serving \(p_{k}\
otin principal_{r}\). The problem can be then modeled as follows, and Table 1 provides the summaries of the parameters and variable notations.