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
    PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
    Pages184-196
    Number of pages13
    Volume30
    ISBN (Electronic)9783939897781
    DOIs
    StatePublished - Jan 1 2015
    Event32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015 - Garching, Germany
    Duration: Mar 4 2015Mar 7 2015

    Other

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

    Fingerprint

    Polynomials
    Combinatorial optimization
    Approximation algorithms

    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 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.

    32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015. Vol. 30 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2015. p. 184-196.

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

    Bus, N, Garg, S, Mustafa, NH & Ray, S 2015, Improved local search for geometric hitting set. in 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
    Bus N, Garg S, Mustafa NH, Ray S. Improved local search for geometric hitting set. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015. Vol. 30. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2015. p. 184-196 https://doi.org/10.4230/LIPIcs.STACS.2015.184
    Bus, Norbert ; Garg, Shashwat ; Mustafa, Nabil H. ; Ray, Saurabh. / Improved local search for geometric hitting set. 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015. Vol. 30 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2015. pp. 184-196
    @inproceedings{0b2eaa669cb94432abdea6602b5ec053,
    title = "Improved local search for geometric hitting set",
    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 {\~O}(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.",
    keywords = "Delaunay triangulation, Disks, Geometric algorithms, Hitting sets, Local search",
    author = "Norbert Bus and Shashwat Garg and Mustafa, {Nabil H.} and Saurabh Ray",
    year = "2015",
    month = "1",
    day = "1",
    doi = "10.4230/LIPIcs.STACS.2015.184",
    language = "English (US)",
    volume = "30",
    pages = "184--196",
    booktitle = "32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015",
    publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

    }

    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 -