Skip to main content
SpringerLink
Log in

Parameterized Complexity of Geometric Covering Problems Having Conflicts

  • Original Research
  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The input for the Geometric Coverage problem consists of a pair \(\varSigma =(P,\mathcal {R})\), where P is a set of points in \({\mathbb {R}}^d\) and \(\mathcal {R}\) is a set of subsets of P defined by the intersection of P with some geometric objects in \({\mathbb {R}}^d\). Motivated by what are called choice problems in geometry, we consider a variation of the Geometric Coverage problem where there are conflicts on the covering objects that precludes some objects from being part of the solution if some others are in the solution. As our first contribution, we propose two natural models in which the conflict relations are given: (a) by a graph on the covering objects, and (b) by a representable matroid on the covering objects. Our main result is that as long as the conflict graph has bounded arboricity there is a parameterized reduction to the conflict-free version. As a consequence, we have the following results when the conflict graph has bounded arboricity. (1) If the Geometric Coverage problem is fixed parameter tractable (FPT), then so is the conflict free version. (2) If the Geometric Coverage problem admits a factor \(\alpha \)-approximation, then the conflict free version admits a factor \(\alpha \)-approximation algorithm running in FPT time. As a corollary to our main result we get a plethora of approximation algorithms that run in FPT time. Our other results include an FPT algorithm and a hardness result for conflict-free version of Covering Points by Intervals. The FPT algorithm is for the case when the conflicts are given by a representable matroid. We prove that conflict-free version of Covering Points by Intervals does not admit an FPT algorithm, unless FPT =W[1], for the family of conflict graphs for which the Independent Set problem is W[1]-hard.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. We refer readers to page number 15, paragraph titled “Capping the objective function at\(k +1\)” in [32] for the explanation of capping the objective function to \(k+1\) in the parameterized approximation.

References

  1. Arkin, E.M., Banik, A., Carmi, P., Citovsky, G., Katz, M.J., Mitchell, J.S.B., Simakov, M.: Choice is hard. In: Proc. 26th Internat. Sympos. Algorithms and Computation, ISAAC 2015, pp. 318–328 (2015). https://doi.org/10.1007/978-3-662-48971-0_28

    Chapter  Google Scholar 

  2. Arkin, E.M., Banik, A., Carmi, P., Citovsky, G., Katz, M.J., Mitchell, J.S.B., Simakov, M.: Conflict-free covering. In: Proc. 27th Canadian Conf. on Comput. Geom., CCCG 2015 (2015)

  3. Arkin, E.M., Díaz-Báñez, J.M., Hurtado, F., Kumar, P., Mitchell, J.S.B., Palop, B., Pérez-Lantero, P., Saumell, M., Silveira, R.I.: Bichromatic 2-center of pairs of points. Comput. Geom. 48(2), 94–107 (2015). https://doi.org/10.1016/j.comgeo.2014.08.004

    Article  MathSciNet  MATH  Google Scholar 

  4. Arkin, E.M., Hassin, R.: Minimum-diameter covering problems. Networks 36(3), 147–155 (2000)

    Article  MathSciNet  Google Scholar 

  5. Aronov, B., de Berg, M., Ezra, E., Sharir, M.: Improved bounds for the union of locally fat objects in the plane. SIAM J. Comput. 43(2), 543–572 (2014)

    Article  MathSciNet  Google Scholar 

  6. Banik, A., Panolan, F., Raman, V., Sahlot, V.: Fréchet distance between a line and avatar point set. Algorithmica 80(9), 2616–2636 (2018). https://doi.org/10.1007/s00453-017-0352-y

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonnet, É., Miltzow, T.: An approximation algorithm for the art gallery problem. In: 33rd International Symposium on Computational Geometry, SoCG 2017, 4–7 July 2017, Brisbane, Australia, pp. 20:1–20:15 (2017). https://doi.org/10.4230/LIPIcs.SoCG.2017.20

  8. Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995)

    Article  MathSciNet  Google Scholar 

  9. Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46(2), 178–189 (2003)

    Article  MathSciNet  Google Scholar 

  10. Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)

    Article  MathSciNet  Google Scholar 

  11. Consuegra, M.E., Narasimhan, G.: Geometric avatar problems. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2013, LIPIcs, vol. 24, pp. 389–400 (2013). https://doi.org/10.4230/LIPIcs.FSTTCS.2013.389

  12. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  Google Scholar 

  13. Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM 52(6), 866–893 (2005). https://doi.org/10.1145/1101821.1101823

    Article  MathSciNet  MATH  Google Scholar 

  14. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Berlin (2012)

    Google Scholar 

  15. Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)

    Book  Google Scholar 

  16. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  17. Edmonds, J.: How to Think About Algorithms. Cambridge University Press, New York (2008)

    Book  Google Scholar 

  18. Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F.A., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. Algorithmica 52(2), 167–176 (2008). https://doi.org/10.1007/s00453-007-9146-y

    Article  MathSciNet  MATH  Google Scholar 

  19. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)

    Google Scholar 

  20. Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009). https://doi.org/10.1007/s00453-007-9133-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63(4), 29 (2016)

    Article  MathSciNet  Google Scholar 

  22. Gabow, H.N., Maheshwari, S.N., Osterweil, L.J.: On two problems in the generation of program test paths. IEEE Trans. Softw. Eng. 2(3), 227–231 (1976)

    Article  MathSciNet  Google Scholar 

  23. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica 7(5&6), 465–497 (1992)

    Article  MathSciNet  Google Scholar 

  24. Har-Peled, S., Quanrud, K.: Approximation algorithms for low-density graphs (2015). CoRR arXiv:1501.00721

  25. Har-Peled, S., Quanrud, K.: Approximation algorithms for polynomial-expansion and low-density graphs. In: Algorithms—ESA 2015—23rd Annual European Symposium, Patras, Greece, 14–16 September 2015, Proceedings, vol. 9294, pp. 717–728. Springer (2015)

  26. Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of a symposium on the Complexity of Computer Computations, pp. 85–103 (1972)

    Chapter  Google Scholar 

  27. Kratsch, S., Philip, G., Ray, S.: Point line cover: the easy kernel is essentially tight. ACM Trans. Algorithms 12(3), 40:1–40:16 (2016). https://doi.org/10.1145/2832912

    Article  MathSciNet  MATH  Google Scholar 

  28. Krohn, E., Gibson, M., Kanade, G., Varadarajan, K.R.: Guarding terrains via local search. JoCG 5(1), 168–178 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33(4), 717–729 (2005)

    Article  MathSciNet  Google Scholar 

  30. Liu, C., Veeraraghavan, K., Iyengar, V.: Thermal-aware test scheduling and hot spot temperature minimization for core-based systems. In: 20th IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems (DFT’05), pp. 552–560. IEEE (2005)

  31. Lokshtanov, D., Misra, P., Panolan, F., Saurabh, S.: Deterministic truncation of linear matroids. ACM Trans. Algorithms 14(2), 14:1–14:20 (2018). https://doi.org/10.1145/3170444

    Article  MathSciNet  MATH  Google Scholar 

  32. Lokshtanov, D., Panolan, F., Ramanujan, M.S., Saurabh, S.: Lossy kernelization. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, 19-23 June 2017, pp. 224–237 (2017). https://doi.org/10.1145/3055399.3055456

  33. Marx, D.: Efficient approximation schemes for geometric problems? In: ESA, pp. 448–459. Springer (2005)

  34. Marx, D.: A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410(44), 4471–4479 (2009)

    Article  MathSciNet  Google Scholar 

  35. Mustafa, N.H., Raman, R., Ray, S.: Settling the APX-hardness status for geometric set cover. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, pp. 541–550. IEEE Computer Society (2014)

  36. Naor, M., Schulman, J.L., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191 (1995)

  37. Oxley, J.G.: Matroid Theory, vol. 3. Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  38. Raman, V., Saurabh, S.: Short cycles make W-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles. Algorithmica 52(2), 203–225 (2008)

    Article  MathSciNet  Google Scholar 

  39. Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. pp. 887–898. ACM (2012)

Download references

Acknowledgements

Funding was provided by FP7 Ideas: European Research Council (Grant No. 306992).

Author information

Authors and Affiliations

  1. National Institute of Science Education and Research, Bhubaneswar, India

    Aritra Banik

  2. Department of Informatics, University of Bergen, Bergen, Norway

    Fahad Panolan

  3. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Venkatesh Raman & Saket Saurabh

  4. Indian Institute of Technology, Jodhpur, India

    Vibha Sahlot

Authors
  1. Aritra Banik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fahad Panolan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Venkatesh Raman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vibha Sahlot
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Saket Saurabh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saket Saurabh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this paper appeared in the proceedings of WADS 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Banik, A., Panolan, F., Raman, V. et al. Parameterized Complexity of Geometric Covering Problems Having Conflicts. Algorithmica 82, 1–19 (2020). https://doi.org/10.1007/s00453-019-00600-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-019-00600-w

Keywords

Search

Navigation