On the Complexity of Computing a Fastest Temporal Path in Interval Temporal Graphs

TL;DR

This work investigates the computational complexity of fastest temporal path problems in interval temporal graphs, contrasting them with the point-temporal model. It establishes a conditional hardness gap via subcubic reductions from negative triangle and triangle detection, showing that under standard conjectures no near-linear-time algorithm can solve fastest temporal paths in the interval model (undirected, constant delays) for general inputs, while also giving a near-linear-time combinatorial algorithm for the special case of zero-delay undirected graphs by computing a complete st-profile. The results illuminate a clear separation between point and interval models and between fastest and shortest path problems, and they identify a natural, broad class of instances where efficient solutions are achievable. Collectively, they advance our understanding of how temporal representation and delay structures impact the tractability of core graph problems with time as a resource.

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, while in the more general interval model, edges can be present over multiple 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 complexity issues. 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. Interestingly, we show that near-linear time is possible in the interval model when restricted to all delays being zero, i.e. traversing an edge is instantaneous.

Figures (2)

• Figure 1: A schematic view of the reduction from Negative Triangle Detection to Fastest Temporal Path.
• Figure 2: A temporal path in $H_G$ corresponding to a triangle in $G$.

Theorems & Definitions (17)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Proposition 1
• proof : Proof of Proposition \ref{['prop:profile']}
• Claim 1
• Claim 2