Minimum-cost coverage of point sets by disks

Abstract

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&#8712;R</i><sup>2</sup> at the smallest possible cost. We consider cost functions of the form &#917;<i><sub>j</sub>f(r<sub>j</sub>)</i>, where <i>f(r)=r</i><sup>&#945;</sup> is the cost of transmission to radius <i>r</i>. Special cases arise for &#945;=1 (sum of radii) and &#945;=2 (total area); power consumption models in wireless network design often use an exponent &#945;&gt;2. Different scenarios arise according to possible restrictions on the transmission centers <i>t<sub>j</sub></i>, 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 <i>t<sub>j</sub></i> on a given line in order to cover demand points <i>Y&#8712;R</i><sup>2</sup>; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover <i>Y</i>; (c) a proof of NP-hardness for a discrete set of transmission points in <i>R<sup>2</sup></i> and any fixed &#945;&gt;1; and (d) a polynomial-time approximation scheme for the problem of computing a <i>minimum cost covering tour</i> (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.

DOI: 10.1145/1137856.1137922

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@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 S. B. Mitchell and Kim Whittlesey}, booktitle={Symposium on Computational Geometry}, year={2006} }