TY - GEN

T1 - Geometric hitting sets for disks

T2 - 23rd European Symposium on Algorithms, ESA 2015

AU - Bus, Norbert

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a setD of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects inD. In 1994, Bronniman and Goodrich [6] made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation ratios. Finally, recently Agarwal-Pan [5] showed that their scheme can be implemented in near-linear time for disks in the plane. This current state-of-the-art is lacking in three ways. First, the constants in current ϵ-net constructions are large, so the approximation factor ends up being more than 40. Second, the algorithm uses sophisticated geometric tools and data structures with large resulting constants. Third, these have resulted in a lack of available software for fast computation of small hitting-sets. In this paper, we make progress on all three of these barriers: i) we prove improved bounds on sizes of ϵ-nets, ii) design hitting-set algorithms without the use of these datastructures and finally, iii) present dnet, a public source-code module that incorporates both of these improvements to compute small-sized hitting sets and ϵ-nets efficiently in practice.

AB - The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a setD of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects inD. In 1994, Bronniman and Goodrich [6] made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation ratios. Finally, recently Agarwal-Pan [5] showed that their scheme can be implemented in near-linear time for disks in the plane. This current state-of-the-art is lacking in three ways. First, the constants in current ϵ-net constructions are large, so the approximation factor ends up being more than 40. Second, the algorithm uses sophisticated geometric tools and data structures with large resulting constants. Third, these have resulted in a lack of available software for fast computation of small hitting-sets. In this paper, we make progress on all three of these barriers: i) we prove improved bounds on sizes of ϵ-nets, ii) design hitting-set algorithms without the use of these datastructures and finally, iii) present dnet, a public source-code module that incorporates both of these improvements to compute small-sized hitting sets and ϵ-nets efficiently in practice.

KW - Approximation Algorithms

KW - Computational Geometry

KW - Geometric Hitting Sets

UR - http://www.scopus.com/inward/record.url?scp=84945581174&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945581174&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-48350-3_75

DO - 10.1007/978-3-662-48350-3_75

M3 - Conference contribution

AN - SCOPUS:84945581174

SN - 9783662483497

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 903

EP - 914

BT - Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings

A2 - Bansal, Nikhil

A2 - Finocchi, Irene

PB - Springer Verlag

Y2 - 14 September 2015 through 16 September 2015

ER -