# Minimum-cost coverage of point sets by disks

@inproceedings{Alt2006MinimumcostCO, title={Minimum-cost coverage of point sets by disks}, author={Helmut Alt and Esther M. Arkin and Herv{\'e} Br{\"o}nnimann and Jeff Erickson and S{\'a}ndor P. Fekete and Christian Knauer and Jonathan Lenchner and Joseph B. M. Mitchell and Kim Whittlesey}, booktitle={SCG '06}, year={2006} }

We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t<sub>j</sub>)</i> and radii <i>(r<sub>j</sub>)</i> that cover a given set of demand points <i>Y∈R</i><sup>2</sup> at the smallest possible cost. We consider cost functions of the form Ε<i><sub>j</sub>f(r<sub>j</sub>)</i>, where <i>f(r)=r</i><sup>α</sup> is the cost of transmission to radius <i>r</i>. Special cases arise for α=1 (sum of radii) and α=2…

## 100 Citations

A note on minimum-sum coverage by aligned disks

- Mathematics, Computer ScienceInf. Process. Lett.
- 2013

This paper considers a facility location problem to find a minimum-sum coverage of n points by disks centered at a fixed line and presents a faster algorithm that runs in O(n^2logn) time for any @a>1 and any L"p-metric.

On the Coverage of Points in the Plane by Disks Centered at a Line

- Physics, Computer ScienceCCCG
- 2018

This paper presents a new algorithm that runs in O(n) time for any α ≥ 1 in any fixed Lp metric and gives algorithms for two variations of the 1D problem where all points of P are in L.

Multi Cover of a Polygon Minimizing the Sum of Areas

- Mathematics, Computer ScienceWALCOM
- 2011

A geometric optimization problem that arises in sensor network design, where one has to cover a given set of n points, X, by disks centered at points of Y, is considered, and a polynomial-time c2-approximation algorithm is presented, where c2 = c2(k) is a constant.

Algorithms for the Line-Constrained Disk Coverage and Related Problems

- Computer ScienceWADS
- 2021

This paper considers a line-constrained version of the disk coverage problem, in which all disks are centered on a line L (while points of P can be anywhere in the plane), and presents an O((m+n) log(m + n) + κ logm) time algorithm for the problem, where κ is the number of pairs of disks that intersect.

Multi Cover of a Polygon Minimizing the Sum of Areas

- Mathematics, Computer ScienceInt. J. Comput. Geom. Appl.
- 2011

This paper presents a polynomial-time c2-approximation algorithm for this problem, where c2 = c2(k) is a constant, and a discrete version, where one has to cover a given set of n points, X, by disks centered at points of Y, arises as a subproblem.

A note on multicovering with disks

- Mathematics, Computer ScienceComput. Geom.
- 2013

An algorithm to find a minimum weight radius assignment that covers each point in P at least K times is presented, where k"m"a"x is the maximum covering requirement of a point, and a (3^@aK+@e)-approximation algorithm for Polygon Disk Multicover is presented.

On Metric Multi-Covering Problems

- Mathematics, Computer Science
- 2016

This article presents an efficient algorithm that reduces the MMC problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$.

Cheap or Flexible Sensor Coverage

- Computer ScienceDCOSS
- 2009

This work considers dual classes of geometric coverage problems, in which disks, corresponding to coverage regions of sensors, are used to cover a region or set of points in the plane, and shows for most settings how to assign either (near-)optimal radius values or allowable amounts of placement error.

Fault-Tolerant Covering Problems in Metric Spaces

- Computer Science, MathematicsAlgorithmica
- 2021

The first constant approximations for these fault-tolerant covering problems in metric spaces are presented, based on the following paradigm: an efficient algorithm is presented that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1.

Computing k-Centers On a Line

- Mathematics, Computer ScienceArXiv
- 2009

In this paper we consider several instances of the k-center on a line problem where the goal is, given a set of points S in the plane and a parameter k >= 1, to find k disks with centers on a line l…

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