Envy\mbox{-}free Trip Planning in Group Trip Planning Query Problem

Abstract: In recent times, Group Trip Planning Query (henceforth referred to as GTP Query) is one of the well\mbox{-}studied problems in Spatial Databases. The inputs to the problem are a road network where the vertices represent the Point-of-Interests (mentioned as POIs henceforth) and they are grouped into different categories, edges represent the road segments, and edge weight represents the distance and a group of users along with their source and destination location. This problem asks to return one POI from every category such that the aggregated distance traveled by the group is minimized. As the objective is to minimize the aggregated distance, the existing solution methodologies do not consider the individual distances traveled by the group members. To address this issue, we introduce and study the \textsc{Envy Free Group Trip Planning Query} Problem. Along with the inputs of the GTP Query Problem, in this variant, we also have a threshold distance $D$ such that aggregated distance traveled by the group is minimized and for any member pairs the difference between their individual distance traveled is less than equal to $D$. However, it may so happen that a given $D$ value no such set POIs are found. To tackle this issue, we introduce the surrogate problem \textsc{Envy Free Group Trip Planning Query with Minimum Additional Distance} Problem which asks what is the minimum distance to be added with $D$ to obtain at least one solution. For these problems, we design efficient solution approaches and experiment with real-world datasets. From the experiments, we observe that the proposed solution approaches lead to less aggregated distance compared to baseline methods with reasonable computational overhead.