Advertisement

Covering and Packing of Triangles Intersecting a Straight Line

  • Supantha PanditEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

Abstract

We study the four geometric optimization problems: Open image in new window , Open image in new window , Open image in new window , and Open image in new window with Open image in new window (a triangle is a right-triangle whose base is parallel to the x-axis, perpendicular is parallel to the y-axis, and the slope of the hypotenuse is \(-1\)). The input triangles are constrained to be intersecting a Open image in new window . The straight line can either be a Open image in new window or an Open image in new window line (a line whose slope is \(-1\)). A right-triangle is said to be a Open image in new window , if the length of both its base and perpendicular is \(\lambda \). For \(1\)-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are \(\mathsf {NP}\)-hard, whereas the piercing set and independent set problems are in \(\mathsf {P}\). The same results hold for \(1\)-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are \(\mathsf {NP}\)-hard. Finally, we give an \(n^{O(\lceil \log c\rceil +1)}\) time exact algorithm for the independent set problem with \(\lambda \)-right-triangles intersecting a straight line such that \(\lambda \) takes more than one value from [1, c], for some integer c. We also present \(O(n^2)\) time dynamic programming algorithms for the independent set problem with \(1\)-right-triangles where the triangles intersect a horizontal line and an inclined line.

Keywords

Set cover Hitting set Piercing set Independent set Horizontal line Inclined line Diagonal line \(\mathsf {NP}\)-hard Right triangles Dynamic programming. 

References

  1. 1.
    Catanzaro, D., et al.: Max point-tolerance graphs. Discret. Appl. Math. 216, 84–97 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chepoi, V., Felsner, S.: Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve. Comput. Geom. 46(9), 1036–1041 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Correa, J., Feuilloley, L., Pérez-Lantero, P., Soto, J.A.: Independent and hitting sets of rectangles intersecting a diagonal line: algorithms and complexity. Discret. Comput. Geom. 53(2), 344–365 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Das, G.K., De, M., Kolay, S., Nandy, S.C., Sur-Kolay, S.: Approximation algorithms for maximum independent set of a unit disk graph. Inf. Process. Lett. 115(3), 439–446 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fraser, R., López-Ortiz, A.: The within-strip discrete unit disk cover problem. Theor. Comput. Sci. 674, 99–115 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discret. Math. 5(3), 422–427 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kratochvíl, J., Nešetřil, J.: INDEPENDENT SET and CLIQUE problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae 031(1), 85–93 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lubiw, A.: A weighted min-max relation for intervals. J. Comb. Theory Ser. B 53(2), 151–172 (1991)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mudgal, A., Pandit, S.: Covering, hitting, piercing and packing rectangles intersecting an inclined line. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 126–137. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-26626-8_10CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Stony Brook UniversityStony BrookUSA

Personalised recommendations