Maximizing reachability in a temporal graph obtained by assigning starting times to a collection of walks

In a temporal graph, each edge appears and can be traversed at specific points in time. In such a graph, temporal reachability of one node from another is naturally captured by the existence of a temporal path where edges appear in chronological order. Inspired by the optimization of bus/metro/tramway schedules in a public transport network, we consider the problem of turning a collection of walks (called trips) in a directed graph into a temporal graph by assigning a starting time to each trip in order to maximize the reachability among pairs of nodes. Each trip represents the trajectory of a vehicle and its edges must be scheduled one right after another. Setting a starting time to the trip thus forces the appearance time of all its edges. We call such a starting time assignment a trip temporalization. We obtain several results about the complexity of maximizing reachability via trip temporalization. Among them, we show that maximizing reachability via trip temporalization is hard to approximate within a factor n/12$$ \sqrt{n}/12 $$ in an n$$ n $$-vertex digraph, even if we assume that for each pair of nodes, there exists a trip temporalization connecting them. On the positive side, we show that there must exist a trip temporalization connecting a constant fraction of all pairs if we additionally assume symmetry, that is, when the collection of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order.