Minimum-Cost Coverage of Point Sets by Disks

Esther M. Arkin; Herve Broennimann; Jeff Erickson; Sandor P. Fekete; Christian Knauer; Jonathan Lenchner; Joseph S. B. Mitchell; Kim Whittlesey

arXiv · cs/0604008

Minimum-Cost Coverage of Point Sets by Disks

Esther M. Arkin·Herve Broennimann·Jeff Erickson·Sandor P. Fekete·Christian Knauer·Jonathan Lenchner·Joseph S. B. Mitchell·Kim Whittlesey

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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 in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>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 in the plane; (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 the plane and any fixed alpha>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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Esther M. Arkin, Herve Broennimann, Jeff Erickson, Sandor P. Fekete, Christian Knauer, Jonathan Lenchner, Joseph S. B. Mitchell, Kim Whittlesey. 2006-04-04. Minimum-Cost Coverage of Point Sets by Disks. https://arxiv.org/abs/cs/0604008

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