Temporal Vertex Cover with a Sliding Time Window
Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an underlying graph topology, which is subject to discrete changes over time. Given a static underlying graph G, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge of G, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs has focused on the notion of a temporal path and 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 and in transportation networks, we introduce and study two natural temporal extensions of the classical problem Vertex Cover. In our first problem, Temporal Vertex Cover, the aim is to cover every edge at least once during the lifetime of the temporal graph, where an edge can only be covered by one of its endpoints at a time step when it is active. In our second, more pragmatic variation Sliding Window Temporal Vertex Cover, we are also given a natural number Delta, and our aim is to cover every edge at least once at every Delta consecutive time steps. In both cases we wish to minimize the total number of "vertex appearances" that are needed to cover the whole graph. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. In particular, we provide strong hardness results, complemented by various approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions.
Temporal networks
temporal vertex cover
APX-hard
approximation algorithm
Exponential Time Hypothesis
Mathematics of computing~Graph theory
Mathematics of computing~Graph algorithms
148:1-148:14
Regular Paper
Partially supported by NeST initiative of the School of EEE and CS at the University of Liverpool and by the EPSRC Grants EP/P020372/1 and EP/P02002X/1.
https://arxiv.org/abs/1802.07103
Eleni C.
Akrida
Eleni C. Akrida
Department of Computer Science, University of Liverpool, UK
https://orcid.org/0000-0002-1126-1623
George B.
Mertzios
George B. Mertzios
Department of Computer Science, Durham University, UK
https://orcid.org/0000-0001-7182-585X
Paul G.
Spirakis
Paul G. Spirakis
Department of Computer Science, University of Liverpool, UK, Department of Computer Engineering & Informatics, University of Patras, Greece
https://orcid.org/0000-0001-5396-3749
Viktor
Zamaraev
Viktor Zamaraev
Department of Computer Science, Durham University, UK
https://orcid.org/0000-0001-5755-4141
10.4230/LIPIcs.ICALP.2018.148
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Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev
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