Temporal Vertex Cover with a Sliding Time Window

Modern, inherently dynamic systems are usually characterized by a network structure, i.e. anunderlying graph topology, which is subject to discrete changes over time. Given a static underlyinggraphG, a temporal graph can be represented via an assignment of a set of integer time-labelsto every edge ofG, indicating the discrete time steps when this edge is active. While most ofthe recent theoretical research on temporal graphs has focused on the notion of a temporal pathand other “path-related” temporal notions, only few attempts have been made to investigate“non-path” temporal graph problems. In this paper, motivated by applications in sensor andin transportation networks, we introduce and study two natural temporal extensions of theclassical problemVertex Cover.In both cases we wish to minimize the total number of“vertex appearances” that are needed to “cover” the whole temporal graph. In our first problem,Temporal Vertex Cover, the aim is to cover every edge at least once during the lifetimeof the temporal graph, where an edge can be covered by one of its endpoints, only at a timestep when it is active. In our second, more pragmatic variationSliding Window TemporalVertex Cover, we are also given a natural number ∆, and our aim is to cover every edgeatleast onceatevery∆consecutivetime steps. We present a thorough investigation of thecomputational complexity and approximability of these two temporal covering problems.Inparticular, we provide strong hardness results, complemented by various approximation and exactalgorithms. Some of our algorithms are polynomial-time, while others are asymptotically almostoptimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions