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Article

Minimum-cost coverage of point sets by disks

Authors:

Esther M. Arkin,

Hervé Brönnimann,

Sándor P. Fekete,

Christian Knauer,

Jonathan Lenchner,

Joseph S. B. Mitchell,

Kim WhittleseyAuthors Info & Claims

SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry

Pages 449 - 458

https://doi.org/10.1145/1137856.1137922

Published: 05 June 2006 Publication History

Abstract

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∈R² at the smallest possible cost. We consider cost functions of the form Ε_jf(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∈R²; (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 R² 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.

References

[1]

P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Computing Surveys, 30(4):412--458, 1998.

Digital Library

[2]

E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Comput. Geom.: Theory Appl., 17(1--2):25--50, 2000.

Digital Library

[3]

E. M. Arkin and R. Hassin. Approximation algorithms for the geometric covering salesman problem Discrete Applied Math., 55(3):197--218, 1994.

Digital Library

[4]

S. Arora, P. Raghavan, and S. Rao. Approximation schemes for Euclidean k-medians and related problems. In Proc. 30th Annu. ACM Symp. Theory Computing, pages 106--113, 1998.

Digital Library

[5]

C. Bajaj. Proving geometric algorithm non-solvability: An application of factoring polynomials. J. Symbol. Comput., 2(1):99--102, 1986.

Digital Library

[6]

C. Bajaj. The algebraic degree of geometric optimization problems. Discrete & Comput. Geom., 3:177--191, 1988.

Digital Library

[7]

V. Bilò, I. Caragiannis, C. Kaklamanis, and P. Kanellopoulos. Geometric clustering to minimize the sum of cluster sizes. In Proc. 13th European Symp. Algorithms, Vol 3669 of LNCS, pages 460--471, 2005.

Digital Library

[8]

H. Brönnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete & Comput. Geom., 14(4):463--479, 1995.

Digital Library

[9]

M. Charikar and R. Panigrahy. Clustering to minimize the sum of cluster diameters. J. Computer Systems Sci., 68(2):417--441, 2004.

Digital Library

[10]

A. E. F. Clementi, P. Penna, and R. Silvestri. On the power assignment problem in radio networks. Technical Report TR00-054, Electronic Colloquium on Computational Complexity, 2000.

[11]

A. Dumitrescu and J. S. B. Mitchell. Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms, 48(1):135--159, 2003.

Digital Library

[12]

T. Erlebach, K. Jansen, and E. Seidel. Polynomial-time approximation scheme for geometric graphs. SIAM J. Computing, 34(6):1302--1323, 2005.

Digital Library

[13]

S. P. Fekete, R. Klein, and A. Nüchter. Searching with an autonomous robot. In Proc. 20th ACM Annu. Symp. Comput. Geom., pages 449--450, 2004.

Digital Library

[14]

S. P. Fekete, R. Klein, and A. Nüchter. Online searching with an autonomous robot. In Algorithmic Foundations of Robotics VI, Vol 17 of Tracts in Advanced Robotics, pages 139--154. Springer, Berlin, 2005.

[15]

The GAP Group. GAP -- Groups, Algorithms, and Programming, Version 4.4, 2005.

[16]

T. F. Gonzalez. Covering a set of points in multidimensional space. Inf. Proc. Letters, 40(4):181--188, 1991.

[17]

J. Hershberger. Minimizing the sum of diameters efficiently. Comput. Geom.: Theory Appl., 2(2):111--118, 1992.

Digital Library

[18]

D. S. Hochbaum and W. Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130--136, 1985.

Digital Library

[19]

N. Lev-Tov and D. Peleg. Polynomial time approximation schemes for base station coverage with minimum total radii. Computer Networks, 47(4):489--501, 2005.

Digital Library

[20]

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Computing, 28(4):1298--1309, 1999.

Digital Library

[21]

K. Pahlavan and A. H. Levesque. Wireless information networks, Vol 001. Wiley, New York, NY, 2nd ed., 2005.

Digital Library

[22]

J. J. Rotman. Advanced Modern Algebra. Prentice Hall, 2002.

Cited By

Bernardini FBiediger DPineda IKleist LBecker A(2024)Strongly-Connected Minimal-Cost Radio-Networks Among Fixed Terminals Using Mobile Relays and Avoiding No-Transmission Zones2024 IEEE 20th International Conference on Automation Science and Engineering (CASE)10.1109/CASE59546.2024.10711430(2059-2066)Online publication date: 28-Aug-2024
https://doi.org/10.1109/CASE59546.2024.10711430
Liu GWang H(2024)On the line-separable unit-disk coverage and related problemsComputational Geometry: Theory and Applications10.1016/j.comgeo.2024.102122123:COnline publication date: 1-Dec-2024
https://dl.acm.org/doi/10.1016/j.comgeo.2024.102122
Wang WCheng BLi JChen YZhang T(2023)An improved approximation algorithm for the $ k $-prize-collecting minimum power cover problemJournal of Industrial and Management Optimization10.3934/jimo.2023140(0-0)Online publication date: 2023
https://doi.org/10.3934/jimo.2023140
Show More Cited By

Index Terms

Minimum-cost coverage of point sets by disks
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Information

Published In

cover image ACM Conferences

SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry

June 2006

500 pages

ISBN:1595933409

DOI:10.1145/1137856

Program Chairs:
Nina Amenta
University of California at Davis
,
Otfried Cheong
Korea Advanced Institute of Science and Technology

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 05 June 2006

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SoCG06

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SoCG06: 22nd Annual Symposium on Computational Geometry

June 5 - 7, 2006

Arizona, Sedona, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Bernardini FBiediger DPineda IKleist LBecker A(2024)Strongly-Connected Minimal-Cost Radio-Networks Among Fixed Terminals Using Mobile Relays and Avoiding No-Transmission Zones2024 IEEE 20th International Conference on Automation Science and Engineering (CASE)10.1109/CASE59546.2024.10711430(2059-2066)Online publication date: 28-Aug-2024
https://doi.org/10.1109/CASE59546.2024.10711430
Liu GWang H(2024)On the line-separable unit-disk coverage and related problemsComputational Geometry: Theory and Applications10.1016/j.comgeo.2024.102122123:COnline publication date: 1-Dec-2024
https://dl.acm.org/doi/10.1016/j.comgeo.2024.102122
Wang WCheng BLi JChen YZhang T(2023)An improved approximation algorithm for the $ k $-prize-collecting minimum power cover problemJournal of Industrial and Management Optimization10.3934/jimo.2023140(0-0)Online publication date: 2023
https://doi.org/10.3934/jimo.2023140
Fekete SGupta UKeldenich PShah SScheffer C(2023)Worst-Case Optimal Covering of Rectangles by DisksDiscrete & Computational Geometry10.1007/s00454-023-00582-172:3(1232-1283)Online publication date: 6-Oct-2023
https://doi.org/10.1007/s00454-023-00582-1
Cao M(2023)An Approximation Algorithm for Stochastic Power Cover ProblemTheoretical Computer Science10.1007/978-981-99-7743-7_6(96-106)Online publication date: 26-Nov-2023
https://doi.org/10.1007/978-981-99-7743-7_6
Liu GWang H(2023)Geometric Hitting Set for Line-Constrained DisksAlgorithms and Data Structures10.1007/978-3-031-38906-1_38(574-587)Online publication date: 28-Jul-2023
https://doi.org/10.1007/978-3-031-38906-1_38
Zhang QLi WSu QZhang X(2022)A Primal–Dual-Based Power Control Approach for Capacitated Edge ServersSensors10.3390/s2219758222:19(7582)Online publication date: 6-Oct-2022
https://doi.org/10.3390/s22197582
刘晓代涵李思李伟(2022)${\boldsymbol~k}$-prize-collecting minimum power cover problem with submodular penalties on a planeSCIENTIA SINICA Informationis10.1360/SSI-2021-044552:6(947)Online publication date: 6-Jun-2022
https://doi.org/10.1360/SSI-2021-0445
Zhang QLi WSu QZhang X(2022)A Local-Ratio-Based Power Control Approach for Capacitated Access Points in Mobile Edge ComputingProceedings of the 6th International Conference on High Performance Compilation, Computing and Communications10.1145/3546000.3546027(175-182)Online publication date: 23-Jun-2022
https://dl.acm.org/doi/10.1145/3546000.3546027
Pedersen LWang H(2022)Algorithms for the line-constrained disk coverage and related problemsComputational Geometry10.1016/j.comgeo.2022.101883105-106(101883)Online publication date: Aug-2022
https://doi.org/10.1016/j.comgeo.2022.101883
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