Abstract
Temporal graphs (in which edges are active at specified times) are of particular relevance for spreading processes on graphs, e.g. the spread of disease or dissemination of information. Motivated by real-world applications, modification of static graphs to control this spread has proven a rich topic for previous research. Here, we introduce a new type of modification for temporal graphs: the number of active times for each edge is fixed, but we can change the relative order in which (sets of) edges are active. We investigate the problem of determining an ordering of edges that minimises the maximum number of vertices reachable from any single starting vertex; epidemiologically, this corresponds to the worst-case number of vertices infected in a single disease outbreak. We study two versions of this problem, both of which we show to be NP-hard, and identify cases in which the problem can be solved or approximated efficiently.
Highlights
Temporal graphs have emerged recently as a useful structure for representing realworld situations, and as a rich source of new algorithmic problems [1, 10, 11, 12, 14, 15, 16]
In the more general case of Min-Max Reachability Temporal Ordering, we show that the problem remains NP-complete even on trees and directed acyclic graph (DAG), and is W[1]-hard on trees when parameterised by the vertex cover number
We have shown Min-Max Reachability Temporal Ordering is extremely difficult to solve exactly: it remains intractable even in the special case of pairwise disjoint singleton edge-classes, and even two highly restricted cases (DAGs, and trees parameterised by vertex cover number) which are almost trivial in this singleton case become intractable as soon as larger edge-classes are allowed
Summary
Temporal (or dynamic) graphs have emerged recently as a useful structure for representing realworld situations, and as a rich source of new algorithmic problems [1, 10, 11, 12, 14, 15, 16]. If a disease incursion starts at the farm represented by v, the farms at risk of infection are precisely those in the reachability set of v, and it is natural to try to minimise the worst case number of farms that might be infected It is not clear, how one might realistically remove edges from such a graph: forbidding trade is likely impossible. We may require that particular subsets of edges are all active simultaneously: this corresponds, for example, to a set of contacts that will take place at a particular conference, or the trades that will be made at a specific named livestock market event (e.g. the “Spring Bull Sale”), whenever the event in question is scheduled Scenarios of this kind where the timing of contacts can be controlled by an organisation responsible for scheduling events (e.g. an auctioneer) perhaps represent the most likely real-world application for this rescheduling approach. An analogous result holds for directed graphs if we define the edge-class interaction graph by making i and j adjacent if and only if there is some v ∈ V (G) such that v has an in-edge in Ei and an out-edge in Ej, or vice versa
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