Minimum-Cost Coverage of Point Sets by Disks
<|reference_start|>Minimum-Cost Coverage of Point Sets by Disks: 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.<|reference_end|>
@article{arkin2006minimum-cost,
    title={Minimum-Cost Coverage of Point Sets by Disks},
    author={Esther M. Arkin and Herve Broennimann and Jeff Erickson and Sandor P.
  Fekete and Christian Knauer and Jonathan Lenchner and Joseph S. B. Mitchell
  and Kim Whittlesey},
    journal={arXiv preprint arXiv:cs/0604008},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0604008},
    primaryClass={cs.DS cs.CG}
}