29,281 research outputs found
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.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on
Computational Geometry 200
A Constant-Factor Approximation for Multi-Covering with Disks
We consider variants of the following multi-covering problem with disks. We
are given two point sets (servers) and (clients) in the plane, a
coverage function , and a constant . Centered at each server is a single disk whose radius we are free to
set. The requirement is that each client be covered by at least
of the server disks. The objective function we wish to minimize is
the sum of the -th powers of the disk radii. We present a polynomial
time algorithm for this problem achieving an approximation
Approximate Clustering via Metric Partitioning
In this paper we consider two metric covering/clustering problems -
\textit{Minimum Cost Covering Problem} (MCC) and -clustering. In the MCC
problem, we are given two point sets (clients) and (servers), and a
metric on . We would like to cover the clients by balls centered at
the servers. The objective function to minimize is the sum of the -th
power of the radii of the balls. Here is a parameter of the
problem (but not of a problem instance). MCC is closely related to the
-clustering problem. The main difference between -clustering and MCC is
that in -clustering one needs to select balls to cover the clients.
For any \eps > 0, we describe quasi-polynomial time (1 + \eps)
approximation algorithms for both of the problems. However, in case of
-clustering the algorithm uses (1 + \eps)k balls. Prior to our work, a
and a approximation were achieved by
polynomial-time algorithms for MCC and -clustering, respectively, where is an absolute constant. These two problems are thus interesting examples of
metric covering/clustering problems that admit (1 + \eps)-approximation
(using (1+\eps)k balls in case of -clustering), if one is willing to
settle for quasi-polynomial time. In contrast, for the variant of MCC where
is part of the input, we show under standard assumptions that no
polynomial time algorithm can achieve an approximation factor better than
for .Comment: 19 page
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