Minimum Membership Hitting Sets of Axis Parallel Segments

  • N. S. Narayanaswamy
  • S. M. Dhannya
  • C. Ramya
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10976)


The Minimum Membership Set Cover (MMSC) problem is a well studied variant among set covering problems. We study the dual of MMSC problem which we refer to as Minimum Membership Hitting Set (MMHS) problem. Exact Hitting Set (EHS) problem is a special case of MMHS problem. In this paper, we show that EHS problem for hypergraphs induced by horizontal axis parallel segments intersected by vertical axis parallel segments is \(\mathsf {NP}\)-complete. Our reduction shows that finding a hitting set in which the number of times any set is hit is minimized does not admit a \(2-\epsilon \) approximation. In the case when the horizontal segments are intersected by vertical lines (instead of vertical segments), we give an algorithm to optimally solve the MMHS problem in polynomial time. Clearly, this algorithm solves the EHS problem as well. Yet, we present a combinatorial algorithm for the special case of EHS problem for horizontal segments intersected by vertical lines because it provides interesting pointers to forbidden structures of intervals that have exact hitting sets. We also present partial results on such forbidden structures.


  1. 1.
    Cheilaris, P., Gargano, L., Rescigno, A.A., Smorodinsky, S.: Strong conflict-free coloring for intervals. Algorithmica 70(4), 732–749 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    de Fraysseix, H., de Mendez, P.O., Pach, J.: A left-first search algorithm for planar graphs. Discrete Comput. Geom. 13(3), 459–468 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dom, M., Guo, J., Niedermeier, R., Wernicke, S.: Minimum membership set covering and the consecutive ones property. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 339–350. Springer, Heidelberg (2006). Scholar
  4. 4.
    Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15(3), 835–855 (1965)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ghouilahouri, A.: Programmes lineaires-caracterisation des matrices totalement unimodulaires. C. R. Hebd. Seances Acad. Sci. 254(7), 1192 (1962)Google Scholar
  6. 6.
    Hartman, I.B.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hoffman, A., Kruskal, J.: Integral boundary points of convex polyhedra. In: Kuhn, H., Tucker, A. (eds.) Linear Inequalities and Related Systems, Ann. Math. Study 38, 223–246 (1956)Google Scholar
  8. 8.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer, Boston (1972). Scholar
  9. 9.
    Katz, M.J., Lev-Tov, N., Morgenstern, G.: Conflict-free coloring of points on a line with respect to a set of intervals. Comput. Geom. 45(9), 508–514 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom. 30(2), 197–205 (2005). Special Issue on the 19th European Workshop on Computational GeometryMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kuhn, F., von Rickenbach, P., Wattenhofer, R., Welzl, E., Zollinger, A.: Interference in cellular networks: the minimum membership set cover problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 188–198. Springer, Heidelberg (2005). Scholar
  12. 12.
    McConnell, R.M.: A certifying algorithm for the consecutive-ones property. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, Philadelphia, PA, USA, pp. 768–777. Society for Industrial and Applied Mathematics (2004)Google Scholar
  13. 13.
    Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 11:1–11:29 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • N. S. Narayanaswamy
    • 1
  • S. M. Dhannya
    • 1
  • C. Ramya
    • 1
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia

Personalised recommendations