## 2 Problem Description and Modelling
### Basic Concepts  
Definition 1 (Road network \(G\)[17]): The road network is modelled as a directed graph \(G=(N,E,\text{scenicScore},\text{travelTime})\), where \(N\) is a collection of nodes (intersections and dead ends), \(E\in N\times N\) is a collection of edges, \(\text{scenicScore}\) is the _scenic value of nodes_ and edges in the road network, and \(\text{travelTime}\) is the _travel time_ of edges in the road network.  
Definition 2 (Path \(R\)[17]): At time \(t_{0}\), a path is formed by connecting multiple edges in the road network in turn. A path from the source point \(n_{0}\) to the destination \(n_{k}\) is marked as \(R\big{<}\big{<}e_{0,1},e_{1,2},\dots,e_{k-1,k}\big{>},t_{0}\big{>}\), where \(n_{0}\) and \(n_{k}\) belong to the set \(N\), \(e\) belongs to the set \(E\), \(e_{0,1}\) represents the edge of node \(n_{0}\) to \(n_{1}\), and \(t_{0}\) is the departure time.  
Definition 3 (User preference P): User preference is the preference vector for the POI feature type obtained by mining the historical behaviour data of users. The user's preference is expressed as \(P(u)=(x_{1},x_{2},\dots,x_{m})\), where \(m\) is the embedding of the user preference into \(m\)-dimensional space. In this paper, the method described in [26] is adopted to obtain user preferences from users' historical check-in data, see Section 3.2 for details.  
Definition 4 (User query \(Q\)[17]): The user's query is defined as a quad, expressed as \(Q\langle n_{0},n_{k},t_{0},b\rangle\), where \(n_{0}\) is the starting point, \(n_{k}\) is the destination, \(t_{0}\) is the departure time, and \(b\) is the time budget of user travel.  
Definition 5 (Effective area): The area in the road network that the user may visit when starting from \(n_{0}\) at time \(t_{0}\) and arriving at destination \(n_{k}\) within travel time budget \(b\).  
Definition 6 (Scenic edge): In this paper, the edge with a _scenic score greater than 0_ is _recorded as _scenic edge_.  
Definition 7 (Urgency): The urgency is used to describe the urgency of the user's travel, that is, whether the user is anxious to reach the destination. At time \(t_{0}\), the minimum travel time from starting point \(n_{0}\) to destination \(n_{k}\) is determined. In this case, the smaller the travel time budget \(b\), the more anxious the user is to reach the destination. Therefore, the urgency is defined as the ratio of the shortest travel time between \(n_{0}\) and \(n_{k}\) at time \(t_{0}\) to the time budget and is calculated as shown in Formula (1).  
\[urgency=\frac{\min travelTime(n_{0},n_{k},t_{0})}{b} \tag{1}\]  
where \(mintravelTime(n_{0},n_{k},t_{0})\) represents the shortest travel time from \(n_{0}\) to \(n_{k}\) at time \(t_{0}\).  
Definition 8 (Quality ratio [6]): The quality ratio is used to describe the cost performance of the path. That is, the higher the _scenic score and the shorter the travel time is, the better the path is. Therefore, the quality ratio is defined as the ratio of the _scenic path score to the travel time and is calculated as shown in Formula (2).  
\[qualityRatio(R)=\frac{\text{scenicScore}(R)}{travelTime(R)} \tag{2}\]  
where \(\text{scenicScore}(R)\) and \(travelTime(R)\) are the _scenic score_ and _travel time cost_ of the path, respectively. The higher \(qualityRatio(R)\) is, the better the path.