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Algorithmica

pp 1–19 | Cite as

Parameterized Complexity of Geometric Covering Problems Having Conflicts

  • Aritra Banik
  • Fahad Panolan
  • Venkatesh Raman
  • Vibha Sahlot
  • Saket SaurabhEmail author
Original Research
  • 1 Downloads

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.

Keywords

Conflict problems Geometric Coverage Parameterized Complexity Matroids 

Notes

Acknowledgements

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

References

  1. 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
  2. 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)Google Scholar
  3. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arkin, E.M., Hassin, R.: Minimum-diameter covering problems. Networks 36(3), 147–155 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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. 8.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46(2), 178–189 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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. 12.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  13. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Berlin (2012)Google Scholar
  15. 15.
    Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Edmonds, J.: How to Think About Algorithms. Cambridge University Press, New York (2008)CrossRefzbMATHGoogle Scholar
  18. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
  20. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gabow, H.N., Westermann, H.H.: Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica 7(5&6), 465–497 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Har-Peled, S., Quanrud, K.: Approximation algorithms for low-density graphs (2015). CoRR arXiv:1501.00721
  25. 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)Google Scholar
  26. 26.
    Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of a symposium on the Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  27. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Krohn, E., Gibson, M., Kanade, G., Varadarajan, K.R.: Guarding terrains via local search. JoCG 5(1), 168–178 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33(4), 717–729 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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)Google Scholar
  31. 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 MathSciNetzbMATHGoogle Scholar
  32. 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. 33.
    Marx, D.: Efficient approximation schemes for geometric problems? In: ESA, pp. 448–459. Springer (2005)Google Scholar
  34. 34.
    Marx, D.: A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410(44), 4471–4479 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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)Google Scholar
  36. 36.
    Naor, M., Schulman, J.L., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191 (1995)Google Scholar
  37. 37.
    Oxley, J.G.: Matroid Theory, vol. 3. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  38. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. pp. 887–898. ACM (2012)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Institute of Science Education and ResearchBhubaneswarIndia
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.The Institute of Mathematical Sciences, HBNIChennaiIndia
  4. 4.Indian Institute of TechnologyJodhpurIndia

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