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Unique Covering Problems with Geometric Sets

  • Pradeesha Ashok
  • Sudeshna Kolay
  • Neeldhara MisraEmail author
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9198)

Abstract

The Exact Cover problem takes a universe U of n elements, a family \(\mathcal F \) of m subsets of U and a positive integer k, and decides whether there exists a subfamily(set cover) \(\mathcal F '\) of size at most k such that each element is covered by exactly one set. The Unique Cover problem also takes the same input and decides whether there is a subfamily \(\mathcal F ' \subseteq \mathcal F \) such that at least k of the elements \(\mathcal F '\) covers are covered uniquely(by exactly one set). Both these problems are known to be NP-complete. In the parameterized setting, when parameterized by k, Exact Cover is W[1]-hard. While Unique Cover is FPT under the same parameter, it is known to not admit a polynomial kernel under standard complexity-theoretic assumptions.

In this paper, we investigate these two problems under the assumption that every set satisfies a given geometric property \(\Pi \). Specifically, we consider the universe to be a set of n points in a real space \({\mathbb R}^d\), d being a positive integer. When \(d = 2\) we consider the problem when \(\Pi \) requires all sets to be unit squares or lines. When \(d >2\), we consider the problem where \(\Pi \) requires all sets to be hyperplanes in \({\mathbb R}^d\). These special versions of the problems are also known to be NP-complete. When parameterizing by k, the Unique Cover problem has a polynomial size kernel for all the above geometric versions. The Exact Cover problem turns out to be W[1]-hard for squares, but FPT for lines and hyperplanes. Further, we also consider the Unique Set Cover problem, which takes the same input and decides whether there is a set cover which covers at least k elements uniquely. To the best of our knowledge, this is a new problem, and we show that it is NP-complete (even for the case of lines). In fact, the problem turns out to be W[1]-hard in the abstract setting, when parameterized by k. However, when we restrict ourselves to the lines and hyperplanes versions, we obtain FPT algorithms.

Keywords

Polynomial Kernel Reduction Rule Input Instance Unique Cover Exact Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Demaine, E.D., Feige, U., Hajiaghayi, M., Salavatipour, M.R.: Combination Can Be Hard: Approximability of the Unique Coverage Problem. SIAM J. Comput. () 38(4), 1464–1483 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Dom, M., Lokshtanov, D., Saurabh, S.: Kernelization Lower Bounds Through Colors and IDs. ACM Trans. Algorithms 11(2), 13:1–13:20 (2014). doi: 10.1145/2650261 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity, p. 530. Springer-Verlag (1999)Google Scholar
  4. 4.
    Flum, J., Grohe, M.: Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). Springer-Verlag New York Inc (2006)Google Scholar
  5. 5.
    Hochbaum, D.S., Maass, W.: Fast approximation algorithms for a nonconvex covering problem. Journal of algorithms 8(3), 305–323 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Karp, R.M.: Reducibility Among Combinatorial Problems. Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  7. 7.
    Langerman, S., Morin, P.: Covering things with things. Discrete & Computational Geometry 33(4), 717–729 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Marx, Dániel: Efficient approximation schemes for geometric problems? In: Brodal, Gerth Stølting, Leonardi, Stefano (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  9. 9.
    Matoušek, J.: Lectures on discrete geometry, vol. 108. Springer, New York (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Megiddo, N., Tamir, A.: On the Complexity of Locating Linear Facilities in the Plane. Oper. Res. Lett. 1(5), 194–197 (1982). doi: 10.1016/0167-6377(82)90039-6 zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Misra, N., Moser, H., Raman, V., Saurabh, S., Sikdar, S.: The Parameterized Complexity of Unique Coverage and Its Variants. Algorithmica, 517–544 (2013)Google Scholar
  12. 12.
    Mustafa, N., Ray, S.: PTAS for geometric hitting set problems via local search. In: Proceedings of the 25th annual symposium on Computational geometry, pp. 17–22. ACM (2009)Google Scholar
  13. 13.
    Vapnik, V.N., Chervonenkis, A.Y.: On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications 16(2), 264–280 (1971)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Pradeesha Ashok
    • 1
  • Sudeshna Kolay
    • 1
  • Neeldhara Misra
    • 2
    Email author
  • Saket Saurabh
    • 1
  1. 1.Institute of Mathematical SciencesChennaiIndia
  2. 2.Indian Institute of ScienceBangaloreIndia

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