Finding Temporal Paths Under Waiting Time Constraints

Abstract

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 $$\varDelta $$ Δ , referred to as $$\varDelta $$ Δ -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.

Pattern detection in large temporal graphs using algebraic fingerprints

“…Many variants of the path problems are known to be solvable in polynomial-time [51,52]. Surprisingly, a simple variant to check the existence of a temporal path with waiting time constraints was shown to be NP-complete by Casteigts et al [12], more strongly, they proved that the problem is W[1]-hard. A known variant of the temporal-path problem is finding top-k shortest paths, which not only asks us to find a shortest path, but also the next k − 1 shortest paths -which may be longer than the shortest path [26].…”

Temporal Connectivity: Coping with Foreseen and Unforeseen Delays

“…However, this situation is not unprecedented. For example, deciding whether a temporal path under waiting time constraints exists (∆-Restless Temporal Path) is NP-complete [6], while finding temporal walks under waiting time constraints can be done in polynomial time [2]. Similarly, counting foremost temporal paths is #P-hard [22], while counting of foremost temporal walks can be done in polynomial time [23].…”

“…Computation of temporal paths and walks has already been studied intensively [3,24], including specialized settings that are novel to temporal graphs: For example, Bentert et al [2] and Casteigts et al [6] studied temporal walks and paths that are only allowed to have limited waiting time at any vertex.…”

Counting Temporal Paths

“…The temporal setting also offers room for new natural temporal path variants that do not have an analogue in the non-temporal setting. Casteigts et al [19] study the problem of finding restless temporal paths that dwell a upper-bounded number of time steps in each vertex, while Füchsle et al [37] study the problem of finding delay-robust routes in a temporal graph (intuitively, temporal paths that are robust with respect to edge delays); both problems turn out to be NP-hard.…”