Optimizing Reachability Sets in Temporal Graphs by Delaying

A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. The questions about reachability sets are motivated by numerous applications of temporal graphs in network epidemiology, which aim to minimise the spread of infection, and scheduling problems in supply networks in manufacturing with the opposite objectives of maximising coverage and productivity. We introduce control mechanisms for reachability sets that are based on two natural operations of delaying. The first operation, termed merging, is global and batches together consecutive time labels into a single time label in the whole network simultaneously. This corresponds to postponing all events until a particular time. The second, imposes independent delays on the time labels of every edge of the graph. We provide a thorough investigation of the computational complexity of different objectives related to reachability sets when these operations are used. For the merging operation, i.e. global lockdown effect, we prove NP-hardness results for several minimization and maximization reachability objectives, even for very simple graph structures. For the second operation, independent delays, we prove that the minimization problems are NP-hard when the number of allowed delays is bounded. We complement this with a polynomial-time algorithm for minimising the reachability set in case of unbounded delays.