Abstract
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.
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Notes
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A ray is a line with one endpoint and extends infinitely in the other direction.
References
Cheilaris, P., Gargano, L., Rescigno, A.A., Smorodinsky, S.: Strong conflict-free coloring for intervals. Algorithmica 70(4), 732–749 (2014)
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)
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). https://doi.org/10.1007/11785293_32
Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15(3), 835–855 (1965)
Ghouilahouri, A.: Programmes lineaires-caracterisation des matrices totalement unimodulaires. C. R. Hebd. Seances Acad. Sci. 254(7), 1192 (1962)
Hartman, I.B.-A., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)
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)
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). https://doi.org/10.1007/978-1-4684-2001-2_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)
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 Geometry
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). https://doi.org/10.1007/11533719_21
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)
Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 11:1–11:29 (2008)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)
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Narayanaswamy, N.S., Dhannya, S.M., Ramya, C. (2018). Minimum Membership Hitting Sets of Axis Parallel Segments. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_53
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