Skip to main content
SpringerLink
Log in

Maximum Independent Set on \(B_1\)-VPG Graphs

  • Conference paper
  • First Online:
Book cover Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

Abstract

We present two approximation algorithms for the maximum independent set (MIS) problem over the class of \(B_1\)-VPG graphs and also for the subclass, equilateral \(B_1\)-VPG graphs. The first algorithm is shown to have an approximation guarantee of \(O((\log n)^2)\) whereas the second one is shown to have an approximation guarantee of \(O(\log d)\) where d denotes the ratio \(d_{max}/d_{min}\) and \(d_{max}\) and \(d_{min}\) denote respectively the maximum and minimum length of of any arm in the input L-representation of the graph. No approximation algorithms have been known for the MIS problem for these graph classes before. Also, the NP-completeness of the decision version restricted to unit length equilateral \(B_1\)-VPG graphs is established.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM (JACM) 41(1), 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biedl, T.C., Derka, M.: \(1\)-string \(b_2\)-VPG representation of planar graphs. In: 31st International Symposium on Computational Geometry, SoCG 2015, 22–25 June 2015, Eindhoven, The Netherlands, pp. 141–155 (2015)

    Google Scholar 

  4. Chaplick, S., Cohen, E., Stacho, J.: Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 319–330. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  5. Chalopin, J., Gonçalves, D., Ochem, P.: Planar graphs have 1-string representations. Discrete Comput. Geom. 43(3), 626–647 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen, E., Golumbic, M.C., Trotter, W.T., Wang, R.: Posets and VPG graphs. Order, pp. 1–11 (2015)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  8. Chaplick, S., Kobourov, S.G., Ueckerdt, T.: Equilateral L-contact graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 139–151. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  9. Chaplick, S., Ueckerdt, T.: Planar graphs as VPG-graphs. J. Graph Algorithms Appl. 17(4), 475–494 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Felsner, S., Knauer, K., Mertzios, G.B., Ueckerdt, T.: Intersection graphs of L-shapes and segments in the plane. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 299–310. Springer, Heidelberg (2014)

    Google Scholar 

  11. Fox, J., Pach, J.: Computing the independence number of intersection graphs. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, 23–25 January 2011, San Francisco, California, USA, pp. 1161–1165 (2011)

    Google Scholar 

  12. Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. W. H Freeman and Company, San Francisco (1979)

    MATH  Google Scholar 

  14. Halldórsson, M.M.: Approximating discrete collections via local improvements. In: SODA, vol. 95, pp. 160–169 (1995)

    Google Scholar 

  15. Håstad, J.: Clique is hard to approximate within \(n{^{1-\varepsilon }}\). Acta Mathematica 182, 105–142 (1997)

    Article  MathSciNet  Google Scholar 

  16. Hunt, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP-and PSPACE-hard problems for geometric graphs. J. Algorithms 26(2), 238–274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kratochvíl, J., Nesetril, J.: INDEPENDENT SET and CLIQUE problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae 031(1), 85–93 (1990)

    MathSciNet  MATH  Google Scholar 

  18. Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, 22–24 January 1990, San Francisco, California, pp. 138–148 (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

    Abhiruk Lahiri

  2. Theoretical Computer Science, The Institute of Mathematical Sciences, Chennai, India

    Joydeep Mukherjee & C. R. Subramanian

Authors
  1. Abhiruk Lahiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joydeep Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. R. Subramanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joydeep Mukherjee .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Lahiri, A., Mukherjee, J., Subramanian, C.R. (2015). Maximum Independent Set on \(B_1\)-VPG Graphs. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_46

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-26626-8_46

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26625-1

  • Online ISBN: 978-3-319-26626-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation