Complexity Gaps between Point and Interval Temporal Graphs for some Reachability Problems

Abstract

Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration, delay of transmission. In the point model, edges appear at specific points in time, whereas in the more general interval model, edges can be present over specific time intervals. In both models, the delay for traversing an edge can change with each edge appearance. When time is discrete, the two models are equivalent in the sense that the presence of an edge during an interval is equivalent to a sequence of point-in-time occurrences of the edge. However, this transformation can drastically change the size of the input and has implications for complexity. Indeed, we show a gap between the two models with respect to the complexity of the classical problem of computing a fastest temporal path from a source vertex to a target vertex, i.e. a path where edges can be traversed one after another in time and such that the total duration from source to target is minimized. It can be solved in near-linear time in the point model, while we show that the interval model requires quadratic time under classical assumptions of fine-grained complexity. With respect to linear time, our lower bound implies a factor of the number of vertices, while the best known algorithm has a factor of the number of underlying edges. We also show a similar complexity gap for computing a shortest temporal path, i.e. a temporal path with a minimum number of edges. Here our lower bound matches known upper bounds up to a logarithmic factor. Interestingly, we show that near-linear time for fastest temporal path computation is possible in the interval model when it is restricted to uniform delay zero, i.e., when traversing an edge is instantaneous. However, this special case is not exempt from our lower bound for shortest temporal path computation. These two results should be contrasted with the computation of a foremost temporal path, i.e., a temporal path that arrives as early as possible. It is well known that this computation can be solved in near-linear time in both models. We also show that there is no gap in testing the all-to-all temporal connectivity of a temporal graph. We demonstrate a quadratic lower bound that applies to both the interval and point models and aligns with the existing upper bounds.