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. 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. 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

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

On The Complexity of Maximizing Temporal Reachability via Trip Temporalisation

International audienceWe consider the problem of assigning appearing times to the edges of a digraph in order to maximize the (average) temporal reachability between pairs of nodes. Motivated by the application to public transit networks, where edges cannot be scheduled independently one of another, we consider the setting where the edges are grouped into certain walks (called trips) in the digraph and where assigning the appearing time to the first edge of a trip forces the appearing times of the subsequent edges. In this setting, we show that, quite surprisingly, it is NP-complete to decide whether there exists an assignment of times connecting a given pair of nodes. This result allows us to prove that the problem of maximising the temporal reachability cannot be approximated within a factor better than some polynomial term in the size of the graph. We thus focus on the case where, for each pair of nodes, there exists an assignment of times such that one node is reachable from the other. We call this property strong temporalisability. It is a very natural assumption for the application to public transit networks. On the negative side, the problem of maximising the temporal reachability remains hard to approximate within a factor √ n/12 in that setting. Moreover, we show the existence of collections of trips that are strongly temporalisable but for which any assignment of starting times to the trips connects at most an O(1/ √ n) fraction of all pairs of nodes. On the positive side, we show that there must exist an assignment of times that connects a constant fraction of all pairs in the strongly temporalisable and symmetric case, that is, when the set of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order. Keywords:edge labeling edge scheduled network network optimisation temporal graph temporal path temporal reachability time assignmen

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. 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. 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