### Abstract

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D{script} of geometric objects, compute the minimum-sized subset of P that hits all objects in D{script}. For the case where D{script} is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)- local search and give an (8 + ∈)-approximation algorithm with expected running time Õ(n^{2.34}); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n^{15}) - that too just for unit disks. The techniques and ideas generalize to (4, 3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3, 2) local search gives an 8-approximation and no better1. Similarly (4, 3)-local search gives a 5-approximation for all these problems.

Original language | English (US) |
---|---|

Title of host publication | 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 184-196 |

Number of pages | 13 |

Volume | 30 |

ISBN (Electronic) | 9783939897781 |

DOIs | |

State | Published - Jan 1 2015 |

Event | 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015 - Garching, Germany Duration: Mar 4 2015 → Mar 7 2015 |

### Other

Other | 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015 |
---|---|

Country | Germany |

City | Garching |

Period | 3/4/15 → 3/7/15 |

### Fingerprint

### Keywords

- Delaunay triangulation
- Disks
- Geometric algorithms
- Hitting sets
- Local search

### ASJC Scopus subject areas

- Software

### Cite this

*32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015*(Vol. 30, pp. 184-196). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.STACS.2015.184

**Improved local search for geometric hitting set.** / Bus, Norbert; Garg, Shashwat; Mustafa, Nabil H.; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015.*vol. 30, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 184-196, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, Garching, Germany, 3/4/15. https://doi.org/10.4230/LIPIcs.STACS.2015.184

}

TY - GEN

T1 - Improved local search for geometric hitting set

AU - Bus, Norbert

AU - Garg, Shashwat

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D{script} of geometric objects, compute the minimum-sized subset of P that hits all objects in D{script}. For the case where D{script} is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)- local search and give an (8 + ∈)-approximation algorithm with expected running time Õ(n2.34); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n15) - that too just for unit disks. The techniques and ideas generalize to (4, 3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3, 2) local search gives an 8-approximation and no better1. Similarly (4, 3)-local search gives a 5-approximation for all these problems.

AB - Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D{script} of geometric objects, compute the minimum-sized subset of P that hits all objects in D{script}. For the case where D{script} is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)- local search and give an (8 + ∈)-approximation algorithm with expected running time Õ(n2.34); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n15) - that too just for unit disks. The techniques and ideas generalize to (4, 3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3, 2) local search gives an 8-approximation and no better1. Similarly (4, 3)-local search gives a 5-approximation for all these problems.

KW - Delaunay triangulation

KW - Disks

KW - Geometric algorithms

KW - Hitting sets

KW - Local search

UR - http://www.scopus.com/inward/record.url?scp=84923868175&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923868175&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2015.184

DO - 10.4230/LIPIcs.STACS.2015.184

M3 - Conference contribution

VL - 30

SP - 184

EP - 196

BT - 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -