Polynomial time approximation schemes for base station coverage with minimum total radii

Lev-Tov, Nissan; Peleg, David

doi:10.1016/j.comnet.2004.08.012

Cited by 75 publications

(60 citation statements)

References 4 publications

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“…In the discrete case studied by Lev-Tov and Peleg [19], and Biló et al [7], we give improved results. For the discrete 1D problem where Y ⊆ R, we improve their 4-approximation to a linear-time 3-approximation by using a "Closest Center with Growth" (CCG) algorithm, and, as an alternative to the previous O((n + m) 3 ) algorithm [19], we give a near-linear-time 2-approximation that uses a "Greedy Growth" (GG) algorithm.…”

Section: Introductionsupporting

confidence: 48%

“…Without loss of generality, we may assume that X and Y are sorted in the same direction, at an extra cost of O((n + m) log(n + m)). Lev-Tov and Peleg [19] give an O((n + m) 3 ) dynamic programming algorithm for finding an exact solution. Bilò et al [7] show that the problem is solvable in polynomial time for any value of α by reducing it to an integer linear program with a totally unimodular constraint matrix.…”

Section: The One-dimensional Discrete Problem With Linear Costmentioning

confidence: 99%

“…(In the case that k n, the disks in the covering will be of radius 0, and the problem becomes that of computing a TSP tour on Y .) Note that our algorithm gives an alternative to the Lev-Tov and Peleg PTAS [19] for coverage alone.…”

Section: The Case C > 4: a Ptasmentioning

confidence: 99%

“…A quadratic dependence (α = 2) models the total area of the served region, an objective arising in some applications. A linear dependence (α = 1) is sometimes assumed, as in Lev-Tov and Peleg [19], who study the base station coverage problem, minimizing the sum of radii. The linear case is important to study not only in order to simplify the problem and gain insight into the general problem, but also to address those settings in which the linear cost model naturally arises [10,21].…”

Section: Introductionmentioning

confidence: 99%

“…For the discrete 1D problem where Y ⊆ R, we improve their 4-approximation to a linear-time 3-approximation by using a "Closest Center with Growth" (CCG) algorithm, and, as an alternative to the previous O((n + m) 3 ) algorithm [19], we give a near-linear-time 2-approximation that uses a "Greedy Growth" (GG) algorithm. Unfortunately, we cannot extend our proofs to the 1 1 2 D problem.…”

Section: Introductionmentioning

confidence: 99%

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Minimum-cost coverage of point sets by disks

Alt

Arkin

Brönnimann³

et al. 2006

Proceedings of the Twenty-Second Annual Symposium on Computational Geometry

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We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t j ) and radii (r j ) that cover a given set of demand points Y ⊂ R 2 at the smallest possible cost. We consider cost functions of the form ∑ j f (r j ), where f (r) = r α is the cost of transmission to radius r. Special cases arise for α = 1 (sum of radii) and α = 2 (total area); power consumption models in wireless network design often use an exponent α > 2. Different scenarios arise according to possible restrictions on the transmission centers t j , which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t j on a given line in order to cover demand points Y ⊂ R 2 ; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y ; (c) a proof of NP-hardness for a discrete set of transmission points in R 2 and any fixed α > 1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

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Section: Introductionsupporting

confidence: 48%

Section: The One-dimensional Discrete Problem With Linear Costmentioning

confidence: 99%

Section: The Case C > 4: a Ptasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimum-cost coverage of point sets by disks

Alt

Arkin

Brönnimann³

et al. 2006

Proceedings of the Twenty-Second Annual Symposium on Computational Geometry

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Geometric Clustering to Minimize the Sum of Cluster Sizes

Bilò

Caragiannis

Kaklamanis

et al. 2005

Lecture Notes in Computer Science

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Abstract. We study geometric versions of the min-size k-clustering problem, a clustering problem which generalizes clustering to minimize the sum of cluster radii and has important applications. We prove that the problem can be solved in polynomial time when the points to be clustered are located on a line. For Euclidean spaces of higher dimensions, we show that the problem is NP-hard and present polynomial time approximation schemes. The latter result yields an improved approximation algorithm for the related problem of k-clustering to minimize the sum of cluster diameters.

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Partitioning the Nodes of a Graph to Minimize the Sum of Subgraph Radii

Proietti

Widmayer

2006

Algorithms and Computation

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Polynomial time approximation schemes for base station coverage with minimum total radii

Cited by 75 publications

References 4 publications

Minimum-cost coverage of point sets by disks

Minimum-cost coverage of point sets by disks

Geometric Clustering to Minimize the Sum of Cluster Sizes

Partitioning the Nodes of a Graph to Minimize the Sum of Subgraph Radii

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