Advertisement

Approximating Domination on Intersection Graphs of Paths on a Grid

  • Saeed Mehrabi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10787)

Abstract

A graph G is called \({B}_k\)-EPG (resp., \({B}_k\)-VPG), for some constant \(k\ge 0\), if it has a string representation on an axis-parallel grid such that each vertex is a path with at most k bends and two vertices are adjacent in G if and only if the corresponding strings share at least one grid edge (resp., the corresponding strings intersect each other). If two adjacent strings of a B\(_k\)-VPG graph intersect each other exactly once, then the graph is called a one-string B\(_k\)-VPG graph.

In this paper, we study the Minimum Dominating Set problem on \(\textsc {B}_1\text {-}{\textsc {EPG}}\) and \(\textsc {B}_1\text {-}{\textsc {VPG}}\) graphs. We first give an O(1)-approximation algorithm on one-string \(\textsc {B}_1\text {-}{\textsc {VPG}}\) graphs, providing the first constant-factor approximation algorithm for this problem. Moreover, we show that the Minimum Dominating Set problem is APX-hard on \(\textsc {B}_1\text {-}{\textsc {EPG}}\) graphs, ruling out the possibility of a PTAS unless P = NP. Finally, to complement our APX-hardness result, we give constant-factor approximation algorithms for the Minimum Dominating Set problem on two non-trivial subclasses of \(\textsc {B}_1\text {-}{\textsc {EPG}}\) graphs.

References

  1. 1.
    Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theor. Comput. Sci. 237(1–2), 123–134 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex intersection graphs of paths on a grid. J. Graph Algorithms Appl. 16(2), 129–150 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biedl, T., Chan, T.M., Lee, S., Mehrabi, S., Montecchiani, F., Vosoughpour, H.: On guarding orthogonal polygons with sliding cameras. In: Poon, S.-H., Rahman, M.S., Yen, H.-C. (eds.) WALCOM 2017. LNCS, vol. 10167, pp. 54–65. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-53925-6_5CrossRefGoogle Scholar
  4. 4.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discret. Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. ACM Trans. Algorithms 6(2), 40:1–40:18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chalopin, J., Gonçalves, D., Ochem, P.: Planar graphs have 1-string representations. Discret. Comput. Geom. 43(3), 626–647 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chaplick, S., Jelínek, V., Kratochvíl, J., Vyskočil, T.: Bend-bounded path intersection graphs: sausages, noodles, and waffles on a grill. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 274–285. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34611-8_28CrossRefGoogle Scholar
  8. 8.
    Damian, M., Pemmaraju, S.V.: APX-hardness of domination problems in circle graphs. Inf. Process. Lett. 97(6), 231–237 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Damian-Iordache, M., Pemmaraju, S.V.: Constant-factor approximation algorithms for domination problems on circle graphs. ISAAC 1999. LNCS, vol. 1741, pp. 70–82. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-46632-0_8CrossRefGoogle Scholar
  10. 10.
    Damian-Iordache, M., Pemmaraju, S.V.: A (2+\(\varepsilon \))-approximation scheme for minimum domination on circle graphs. J. Algorithms 42(2), 255–276 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Epstein, D., Golumbic, M.C., Morgenstern, G.: Approximation algorithms for B\(_1\)-EPG graphs. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 328–340. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40104-6_29CrossRefGoogle Scholar
  12. 12.
    Heldt, D., Knauer, K.B., Ueckerdt, T.: Edge-intersection graphs of grid paths: the bend-number. Discret. Appl. Math. 167, 144–162 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom. 30(2), 197–205 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lahiri, A., Mukherjee, J., Subramanian, C.R.: Maximum independent set on \(\rm {B}_1\)-VPG graphs. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 633–646. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-26626-8_46CrossRefGoogle Scholar
  15. 15.
    Middendorf, M., Pfeiffer, F.: The max clique problem in classes of string-graphs. Discret. Math. 108(1–3), 365–372 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pergel, M., Rzążewski, P.: On edge intersection graphs of paths with 2 bends. In: Heggernes, P. (ed.) WG 2016. LNCS, vol. 9941, pp. 207–219. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53536-3_18CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations