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.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on
Computational Geometry 200
