Abstract
Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other “path-related” temporal notions, only few attempts have been made to investigate “non-path” temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by 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.
Highlights
Introduction and MotivationA great variety of both modern and traditional networks are inherently dynamic, in the sense that their link availability varies over time
In this paper we present a thorough investigation of the complexity and approximability of the problems Temporal Vertex Cover (TVC) and Sliding Window Temporal Vertex
2 According to the American Public Transportation Association (see http://www.apta.com/resources/ standards/Documents/APTA-RT-VIM-RP-019-03.pdf “developing minimum inspection, maintenance, testing and alignment procedures maintains rail transit trucks in a safe and reliable operating condition”. 148:4 Temporal Vertex Cover with a Sliding Time Window there exists a constant ε ∈ (0, 1) such that Sliding Window Temporal Vertex Cover (SW-TVC) cannot be solved in f (T ) · 2εn·g(∆) time, assuming the Exponential Time Hypothesis (ETH)
Summary
A great variety of both modern and traditional networks are inherently dynamic, in the sense that their link availability varies over time. In this paper we adopt a simple and natural model for time-varying networks which is given with time-labels on the edges of a graph, while the vertex set remains unchanged. This formalism originates in the foundational work of Kempe et al [20]. Several works in the field of sensor networks considered problems of placing sensors to cover a whole area or multiple critical locations, e.g. for reasons of surveillance Such studies usually wish to minimize the number of sensors used or the total energy required [11, 16, 23, 28, 33]. Notice that the above is an application drawn from real-life, as regular checks in roads and trucks are paramount for the correct operation of the transporting sector, according to both the European Commission and the American Public Transportation Association
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