2 research outputs found
Tighter Estimates for epsilon-nets for Disks
The geometric hitting set problem is one of the basic geometric combinatorial
optimization problems: given a set of points, and a set of
geometric objects in the plane, the goal is to compute a small-sized subset of
that hits all objects in . In 1994, Bronniman and Goodrich
made an important connection of this problem to the size of fundamental
combinatorial structures called -nets, showing that small-sized
-nets imply approximation algorithms with correspondingly small
approximation ratios. Very recently, Agarwal and Pan showed that their scheme
can be implemented in near-linear time for disks in the plane. Altogether this
gives -factor approximation algorithms in time for hitting
sets for disks in the plane.
This constant factor depends on the sizes of -nets for disks;
unfortunately, the current state-of-the-art bounds are large -- at least
and most likely larger than . Thus the approximation
factor of the Agarwal and Pan algorithm ends up being more than . The best
lower-bound is , which follows from the Pach-Woeginger construction
for halfspaces in two dimensions. Thus there is a large gap between the
best-known upper and lower bounds. Besides being of independent interest,
finding precise bounds is important since this immediately implies an improved
linear-time algorithm for the hitting-set problem.
The main goal of this paper is to improve the upper-bound to
for disks in the plane. The proof is constructive, giving a simple algorithm
that uses only Delaunay triangulations. We have implemented the algorithm,
which is available as a public open-source module. Experimental results show
that the sizes of -nets for a variety of data-sets is lower, around
A PTAS for the Weighted Unit Disk Cover Problem
We are given a set of weighted unit disks and a set of points in Euclidean
plane. The minimum weight unit disk cover (\UDC) problem asks for a subset of
disks of minimum total weight that covers all given points. \UDC\ is one of the
geometric set cover problems, which have been studied extensively for the past
two decades (for many different geometric range spaces, such as (unit) disks,
halfspaces, rectangles, triangles). It is known that the unweighted \UDC\
problem is NP-hard and admits a polynomial-time approximation scheme (PTAS).
For the weighted \UDC\ problem, several constant approximations have been
developed. However, whether the problem admits a PTAS has been an open
question. In this paper, we answer this question affirmatively by presenting
the first PTAS for \UDC. Our result implies the first PTAS for the minimum
weight dominating set problem in unit disk graphs. Combining with existing
ideas, our result can also be used to obtain the first PTAS for the maxmimum
lifetime coverage problem and an improved constant approximation ratio for the
connected dominating set problem in unit disk graphs.Comment: We fixed several typos in this version. 37 pages. 15 figure