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.

翻译：时序图在建模随时间演化的交互关系时出现。根据描述交互时间特性的参数数量（出现时间、持续时间、传输延迟），时序图通常呈现多种形式。在点模型中，边在特定时间点出现；而在更一般的区间模型中，边可在特定时间区间内持续存在。两种模型中，穿越边的延迟可能随每次边出现而变化。当时间离散时，两种模型在以下意义上是等价的：边在区间内的存在等价于该边在时间点上的序列出现。然而，这种转换可能显著改变输入规模并对复杂度产生影响。我们证明两种模型在计算从源顶点到目标顶点的最快时序路径（即边可随时间依次穿越，且源到目标总时长最小化的路径）这一经典问题的复杂度上存在差异。该问题在点模型中可在近线性时间内求解，而我们证明在细粒度复杂度的经典假设下，区间模型需要二次时间。相对于线性时间，我们的下界意味着顶点数量的倍数因子，而目前已知最优算法具有底层边数量的倍数因子。对于计算最短时序路径（即边数最少的时序路径），我们也展示了类似的复杂度差异。此处我们的下界与已知上界仅相差对数因子。值得注意的是，我们证明当时序图限制在均匀零延迟（即穿越边瞬时完成）时，区间模型中的最快时序路径计算可在近线性时间内实现。但这一特例仍受限于我们针对最短时序路径计算的下界。这两个结果应与最早时序路径（即尽可能早到达的时序路径）的计算形成对比。众所周知，该计算在两种模型中均可在近线性时间内求解。我们还证明在测试时序图的全对全时序连通性时不存在复杂度差异：我们展示的二次时间下界同时适用于区间模型和点模型，且与现有上界一致。