Temporal vertex cover with a sliding time window

Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, Viktor Zamaraev
2019 Journal of computer and system sciences (Print)  
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more » ... statement: c Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis and Viktor Zamaraev; licensed under Creative Commons License CC-BY Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract 20 Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an un-21 derlying graph topology, which is subject to discrete changes over time. Given a static underlying 22 graph G, a temporal graph can be represented via an assignment of a set of integer time-labels 23 to every edge of G, indicating the discrete time steps when this edge is active. While most of 24 the recent theoretical research on temporal graphs has focused on the notion of a temporal path 25 and other "path-related" temporal notions, only few attempts have been made to investigate 26 "non-path" temporal graph problems. In this paper, motivated by applications in sensor and in 27 transportation networks, we introduce and study two natural temporal extensions of the classical 28 problem Vertex Cover. In our first problem, Temporal Vertex Cover, the aim is to cover 29 every edge at least once during the lifetime of the temporal graph, where an edge can only be 30 covered by one of its endpoints at a time step when it is active. In our second, more pragmatic 31 variation Sliding Window Temporal Vertex Cover, we are also given a natural number 32 ∆, and our aim is to cover every edge at least once at every ∆ consecutive time steps. In both 33 cases we wish to minimize the total number of "vertex appearances" that are needed to cover the 34 whole graph. We present a thorough investigation of the computational complexity and approx-35 imability of these two temporal covering problems. In particular, we provide strong hardness 36 results, complemented by various approximation and exact algorithms. Some of our algorithms 37 are polynomial-time, while others are asymptotically almost optimal under the Exponential Time 38 Hypothesis (ETH) and other plausible complexity assumptions. 39 2012 ACM Subject Classification Mathematics of computing → Graph theory • Mathematics 40 of computing → Graph algorithms. E. C. Akrida, G. B. Mertzios, P. G. Spirakis and V. Zamaraev 467:3 then to cover all edges with the minimum total number of such "vertex appearances". On the 85 other hand, in many real-world applications where scalability is important, the lifetime T can 86 be arbitrarily large but the network still needs to remain sufficiently covered. In such cases, 87 as well as in safety-critical systems (e.g. in military applications), it may not be satisfactory 88 enough that an edge is covered just once during the whole lifetime of the network. Instead, 89 every edge must be covered at least once within every small ∆-window of time (for an 90 appropriate value of ∆), regardless of how large the lifetime is; this gives rise to our second 91 optimization problem, namely Sliding Window Temporal Vertex Cover (for short, 92 SW-TVC). Formal definitions of our problems TVC and SW-TVC are given in Section 2. 93 Our two temporal extensions of Vertex Cover are motivated by applications in sensor 94 networks and in transportation networks. In particular, several works in the field of sensor 95 networks considered problems of placing sensors to cover a whole area or multiple critical 96 locations, e.g. for reasons of surveillance. Such studies usually wish to minimize the number 97 of sensors used or the total energy required [11, 16, 23, 28, 33]. Our temporal vertex cover 98
doi:10.1016/j.jcss.2019.08.002 fatcat:m23ibl7ahfcfvfyex2ibbhm5ca