Changing times to optimise reachability in temporal graphs

Temporal graphs (in which edges are active only at specified time steps) are an increasingly important and popular model for a wide variety of natural and social phenomena. We propose a new extension of classical graph modification problems into the temporal setting, and describe several variations on a modification problem in which we assign times to edges so as to maximise or minimise reachability sets within a temporal graph. We give an assortment of complexity results on these problems, showing that they are hard under a variety of restrictions. In particular, if edges can be grouped into classes that must be assigned the same time, then our problem is hard even on directed acyclic graphs when both the reachability target and the classes of edges are of constant size, as well as on an extremely restrictive class of trees. We further show that one version of the problem is W[1]-hard when parameterised by the vertex cover number of the instance graph. In the case that each edge is active at a unique timestep, we identify some very restricted cases in which the problem is solvable in polynomial time; however, list versions of both problems (each edge may only be assigned times from a specified lists) remain NP-complete in this setting even if the graph is of bounded degree and the reachability target is a constant.