Improved local search for geometric hitting set

Norbert Bus, Shashwat Garg, Nabil H. Mustafa, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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(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.

Original languageEnglish (US)
Title of host publication32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015
EditorsErnst W. Mayr, Nicolas Ollinger
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages184-196
Number of pages13
ISBN (Electronic)9783939897781
DOIs
StatePublished - Feb 1 2015
Event32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015 - Garching, Germany
Duration: Mar 4 2015Mar 7 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume30
ISSN (Print)1868-8969

Other

Other32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015
CountryGermany
CityGarching
Period3/4/153/7/15

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Keywords

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

ASJC Scopus subject areas

  • Software

Cite this

Bus, N., Garg, S., Mustafa, N. H., & Ray, S. (2015). Improved local search for geometric hitting set. In E. W. Mayr, & N. Ollinger (Eds.), 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015 (pp. 184-196). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 30). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.STACS.2015.184