Optimizing Reachability Sets in Temporal Graphs by Delaying

Abstract

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 and scheduling problems in supply networks in manufacturing. We introduce control mechanisms for reachability sets that are based on two natural operations of delaying time events. The first operation, termed merging, is global and batches together consecutive time labels 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, we prove NP-hardness results for several minimization and maximization reachability objectives, even for very simple graph structures. For the second operation, 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 the case of unbounded delays.

Counting Temporal Paths

“…Generally, connectivity related problems have received a lot of attention in the temporal setting, ranging from the mentioned temporal path and betweenness computation to finding temporally connected subgraphs [6,20], temporal separation [35,46,50,54,68], temporal graph modification to decrease or increase its connectivity [24,27,28,55], temporal graph exploration [2,13,17,30,31,32], temporal network design [1,51], and others [38,40,48].…”

Temporal Connectivity: Coping with Foreseen and Unforeseen Delays

“…There has been extensive research on many other connectivity-related problems on temporal graphs [4,9,10,11,12,17,18,20,25]. Delays in temporal graphs have been considered in terms of manipulating reachability sets [7,21]. An individual delay operation considered in the mentioned work delays a single time arc and is similar to our notion.…”

“…end(e) / ∈ V ∧ F (end(e), t(e) + λ(e), y, V ∪ {v}) ∧ F (end(e), t(e) + λ(e) + δ, y − 1, V ∪ {v}) , (7) where the empty disjunction evaluates to false. Using ( 5)-( 7), we get the following result by evaluating it in a depth-first-search fashion from the starting configuration.…”

“…Note that since the set V of visited vertices is growing with each move of the traveler, the game cannot run infinitely. Thus, all entries can be computed by means of ( 5)- (7).…”

Temporal Reachability Minimization: Delaying vs. Deleting

“…Application instances for this scenario may be drawn from physical contacts This is the central problem studied in this paper. We remark that technically Deligkas and Potapov [15] formulate the problem slightly differently, allowing delays of up to δ to appear. However, a simple argument can be given to see that this distinction is not significant: Clearly, delaying a time-edge reduces the number of reachable vertices only if the undelayed time-edge could be reached from some source s ∈ S. But when this is the case, increasing the delay of that time-edge can never increase the set of vertices reachable from S, even though it might increase the set of vertices reachable from some s ′ ̸ = s.…”