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 (tj) and radii (rj) that cover a given set of demand points at the smallest possible cost. We consider cost functions of the form
where
is the cost of transmission to radius r. Special cases arise for
(sum of radii) and
(total area) ; power consumption models in wireless network design often use an exponent
.l Different scenarios arise according to possible restrictions on the transmission centers tj, which may be constrained to belong to a given discrete set or to lie on a line, etc.
We obtrain several new results, including (a) exact and approximation algorithms for selecting transmission points tj on a given line in order to cover demand points (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
and any fixed
and (d) a polynomial-time approximatoin scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission coxt for the set of disks and the length of a tour/path that connects the centers of the disks.
By: Helmut Alt, Esther M. Arkin, Hervé Brönnimann, Jeff Erickson, Sándor Fekete, Christian Knauer, Jonathan Lenchner, Joseph S. B. Mitchell, Kim Whittlesey
Published in: RC23831 in 2005
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