The complexity of transitively orienting temporal graphs

In a temporal network with discrete time-labels on its edges, information can only “flow” along sequences of edges with non-decreasing (resp. increasing) time-labels. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. By naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation, and we systematically investigate its algorithmic behavior. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a temporal graph G is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether G is strictly transitively orientable. Additionally we introduce further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.