PTAS for geometric hitting set problems via local search

Mustafa, Nabil H.; Ray, Saurabh

doi:10.1145/1542362.1542367

Cited by 74 publications

(47 citation statements)

References 18 publications

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“…Baker's approach was extended by Eppstein [Epp00] to graphs with bounded local treewidth, and by Grohe [Gro03] to graphs excluding minors. Separators have also played a key role in geometric optimization algorithms, including: (i) PTAS for independent set and (continuous) piercing set for fat objects [Cha03,MR10], (ii) QPTAS for maximum weighted independent sets of polygons [AW13,AW14,Har14], and (iii) QPTAS for Set Cover by pseudodisks [MRR14a], among others. Lastly, Cabello and Gajser [CG14a] develop PTAS's for some of the problems we study in the specific setting of minor-free graphs.…”

Section: Further Related Workmentioning

confidence: 99%

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Har-Peled

Quanrud

2015

Algorithms - ESA 2015

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We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion.We present efficient (1 + ε)-approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS's for these problems are known for subclasses of this graph family. These results have immediate applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow).We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth.

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Section: Further Related Workmentioning

confidence: 99%

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Har-Peled

Quanrud

2015

Algorithms - ESA 2015

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“…In this paper, we use a geometric version of HitSet in which the set of given elements are points in R 2 and the subsets are induced by given disks (i.e., each S ∈ S is the subset of points that can be covered by a given disk). Geometric hitting set admits constant factor approximation algorithms (even PTAS) for many geometric objects (including disks) [2,8,29,32,5]. As mentioned in the introduction, MIN-CSC is a special case of the following group Steiner tree (GST) problem.…”

Section: Preliminariesmentioning

confidence: 99%

Approximation Algorithms for the Connected Sensor Cover Problem

Huang

Shi

2015

Lecture Notes in Computer Science

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Abstract. We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation algorithm, assuming that the ratio between the sensor radius and communication radius is bounded. In Budgeted-CSC problem, our goal is to choose a set of B sensors, such that the number of targets covered by the chosen sensors is maximized and the communication graph is connected. We also obtain a constant approximation under the same assumption.

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“…Recently, Mustafa and Ray [17,18] proposed a PTAS for the discrete geometric hitting set problem. 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%

“…Let us first introduce a result from [18], which is obtained from a classical property on planar graph partition. …”

Section: Lemma 1 (Locality Condition)mentioning

confidence: 99%

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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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PTAS for geometric hitting set problems via local search

Cited by 74 publications

References 18 publications

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

Approximation Algorithms for the Connected Sensor Cover Problem

Wireless coverage with disparate ranges

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