The complexity of transitively orienting temporal graphs

Abstract

In a temporal network with discrete time-labels on its edges, entities and information can only “flow” along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge e = {u, v} with time-label t specifies that “u communicates with v at time t”. This is a symmetric relation between u and v, and it can be interpreted that the information can flow in either direction. 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. More specifically, 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 in various situations. An orientation of a temporal graph is called temporally transitive if, whenever u has a directed edge towards v with time-label t1 and v has a directed edge towards w with time-label t2 ≥ t1, then u also has a directed edge towards w with some time-label t3 ≥ t2. If we just demand that this implication holds whenever t2 > t1, the orientation is called strictly temporally transitive, as it is based on the fact that there is a strict directed temporal path from u to w. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given 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 and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.