The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $\Delta$, referred to as $\Delta$-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

💡 Research Summary

The paper investigates the computational complexity of finding Δ‑restless temporal paths in temporal graphs, where a temporal graph consists of a fixed vertex set and a sequence of edge sets (layers) that evolve over discrete time steps. A Δ‑restless temporal (s, z)‑path is a time‑respecting walk that never revisits a vertex and, crucially, must leave each intermediate vertex within at most Δ time steps after arriving. This models scenarios such as packet routing with bounded buffering time or disease transmission under an SIR model where recovered individuals become immune after a bounded infectious period.

The authors first formalize the problem and its short‑path variant (bounded by a length parameter k). They observe that any Δ‑restless path corresponds to an (s, z)‑path in the underlying static graph, but the converse need not hold; checking whether a static path can be realized as a Δ‑restless temporal path can be done in linear time.

Hardness results are the core of the negative side. Even when the lifetime (the number of time steps) is restricted to three, Restless Temporal Path remains NP‑hard, showing that the difficulty does not vanish in highly constrained temporal windows. More strikingly, the problem is W