Algorithms for Dominating Set in Disk Graphs: Breaking the logn Barrier

Gibson, Matt; Pirwani, Imran A.

doi:10.1007/978-3-642-15775-2_21

Cited by 63 publications

(45 citation statements)

References 18 publications

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“…Disks in R 2 can be mapped to halfspaces in R 3 by a well-known lifting transformation [12], and so we get an O(1)-approximation for weighted hitting set for disks in the plane. Recently, Gibson and Pirwani [16] have applied Varadarajan's technique to the weighted dominating set problem in the intersection graph of a set of disks in R 2 . We can map a disk σ with center (a, b) and radius c to a point p σ = (a, b, c), and a disk σ with center (a , b ) and radius c to a region S σ = {(x, y, z) : (x − a ) 2 + (y − b ) 2 ≤ z + c }, so that the two disks intersect if and only if p σ is covered by S σ .…”

Section: Applicationsmentioning

confidence: 99%

Weighted Capacitated, Priority, and Geometric Set Cover via Improved Quasi-Uniform Sampling

Chan¹,

Grant²,

Könemann³

et al. 2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

135

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The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan's recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC. We also show how to derandomize our algorithm.Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. [8] by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan's sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2 O(log * n) -approximation known prior to our work and matching the O(1)-approximation known for the unweighted variant.

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Section: Applicationsmentioning

confidence: 99%

Weighted Capacitated, Priority, and Geometric Set Cover via Improved Quasi-Uniform Sampling

Chan¹,

Grant²,

Könemann³

et al. 2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

135

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“…As estimated in [35], these constants are at best 20 (A recent result [6] shows that the constant is at most 13). Moreover, there exists a PTAS for unweighted disk cover and minimum dominating set via the local search technique [25,35].…”

Section: Previous Results and Our Contributionmentioning

confidence: 99%

A PTAS for the Weighted Unit Disk Cover Problem

Jin

2015

Lecture Notes in Computer Science

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We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maximum lifetime coverage problem and an improved 4.475-approximation for the connected dominating set problem in unit disk graphs. *

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“…Based on their techniques, Gibson et al [9,10] gave a PTAS for the unweighted case, and 2 O(log * n) -approximation for the weighted case of the problem minimum dominating set in disk intersection graph with arbitrary disk radii.…”

Section: Related Problems and Useful Techniquesmentioning

confidence: 99%

Wireless coverage with disparate ranges

Wan

Wang

2011

Proceedings of the Twelfth ACM International Symposium on Mobile Ad Hoc Networking and Computing

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Coverage has been one of the most fundamental yet challenging issues in wireless networks. Given a set of nodes and a set of disks of disparate radii, the problem Minimum Disk Cover seeks a disk cover of all nodes with minimum cardinality. We present the first polynomial time approximation scheme. We also consider a classical generalization where each input disk is associated with a positive cost, the problem Min-Cost Disk Cover seeks a disk cover of all nodes with minimum total cost. We present a randomized algorithm that can achieve an approximation ratio of 2with high probability, where n is the number of input disks. Another line of this work is exploring the relations between disk cover and an important practical problem which seeks a wireless covering schedule of maximum life subject to an energy budget function. We present two algorithms: Ellipsoid Algorithm (EA) and Price-Directive Algorithm (PDA), and prove that by applying our algorithmic results on disk cover, the approximation ratios for EA and PDA are 2 O(log * n) and (1 + ǫ) 2 O(log * n) respectively.

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Algorithms for Dominating Set in Disk Graphs: Breaking the logn Barrier

Cited by 63 publications

References 18 publications

Weighted Capacitated, Priority, and Geometric Set Cover via Improved Quasi-Uniform Sampling

Weighted Capacitated, Priority, and Geometric Set Cover via Improved Quasi-Uniform Sampling

A PTAS for the Weighted Unit Disk Cover Problem

Wireless coverage with disparate ranges

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