Complexity of Barrier Coverage with Relocatable Sensors in the Plane

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 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 lifetime of the temporal graph, where an edge can be covered by one of its endpoints, only 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 ∆, and our aim is to cover every edge at least once at every ∆ consecutive time steps. 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.

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“…for reasons of surveillance. Such studies usually wish to minimize the number of sensors used or the total energy required [17,25,33,42,48].…”

Section: Introductionmentioning

confidence: 99%

Temporal vertex cover with a sliding time window

Akrida

Mertzios

Spirakis

et al. 2020

Journal of Computer and System Sciences

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“…They also proposed polynomial-time algorithms for several special cases of the problems, e.g., barriers are parallel or perpendicular to each other, and sensors have some constrained movements. In fact, if sensors have different ranges, by an easy reduction from the Partition Problem as in [9], we can show that our problem MBC is NP-hard even for the line-constrained version and m = 2.…”

Section: Related Workmentioning

confidence: 93%

“…Dobrev et al [9] studies several problems on covering multiple barriers in the plane. They showed that these problems are generally NP-hard when sensors have different ranges.…”

Section: Related Workmentioning

confidence: 99%

Algorithms for Covering Multiple Barriers

Wang

2017

Lecture Notes in Computer Science

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In this paper, we consider the problems for covering multiple intervals on a line. Given a set B of m line segments (called "barriers") on a horizontal line L and another set S of n horizontal line segments of the same length in the plane, we want to move all segments of S to L so that their union covers all barriers and the maximum movement of all segments of S is minimized. Previously, an O(n 3 log n)-time algorithm was given for the case m = 1. In this paper, we propose an O(n 2 log n log log n + nm log m)-time algorithm for a more general setting with any m ≥ 1, which also improves the previous work when m = 1. We then consider a line-constrained version of the problem in which the segments of S are all initially on the line L. Previously, an O(n log n)-time algorithm was known for the case m = 1. We present an algorithm of O(m log m + n log m log n) time for any m ≥ 1. These problems may have applications in mobile sensor barrier coverage in wireless sensor networks.

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“…In [10], the complexity of the MinMax and MinSum problems when sensors are initially placed in the plane and are required to relocate to cover parallel or perpendicular barriers is studied. The authors show that while MinMax and MinSum can be solved using dynamic programming in polynomial time if sensors are required to move to the closest point on the barrier, even the feasibility of covering two perpendicular barriers is NP-hard to determine.…”

Section: Related Workmentioning

confidence: 99%

Weak Coverage of a Rectangular Barrier

Dobrev

Kranakis

Kriz̧anc

et al. 2017

Lecture Notes in Computer Science

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Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (MinMax). We give an O(n 3/2 ) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions; in contrast we show that the problem is NP-hard even for sensors of diameter 2 that are initially placed at integer positions. We show that the MinSum problem is solvable in O(n log n) time for homogeneous range sensors in arbitrary initial positions for the Manhattan metric, while it is NP-hard for heterogeneous sensor ranges for both Euclidean and Manhattan metrics. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-hard.

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Complexity of Barrier Coverage with Relocatable Sensors in the Plane

Cited by 12 publications

References 18 publications

Temporal vertex cover with a sliding time window

Temporal vertex cover with a sliding time window

Algorithms for Covering Multiple Barriers

Weak Coverage of a Rectangular Barrier

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