Skip to main content
SpringerLink
Log in

Approximation Algorithms for B 1-EPG Graphs

  • Conference paper
Algorithms and Data Structures (WADS 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8037))

Included in the following conference series:

Abstract

The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths, namely, the class known as B 1-EPG graphs. The motivation for studying these graphs comes from the context of circuit layout problems. It is known that recognizing B 1-EPG graphs is NP-complete, nevertheless, optimization problems when given a set of paths in the grid are of considerable practical interest.

In this paper, we show that the coloring problem and the maximum independent set problem are both NP-complete for B 1-EPG graphs, even when the EPG representation is given. We then provide efficient 4-approximation algorithms for both of these problems, assuming the EPG representation is given. We conclude by noting that the maximum clique problem can be optimally solved in polynomial time for B 1-EPG graphs, even when the EPG representation is not given.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asinowski, A., Ries, B.: Some properties of edge intersection graphs of single bend paths on a grid. Discrete Mathematics 312, 427–440 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asinowski, A., Suk, A.: Edge intersection graphs of systems of grid paths with bounded number of bends. Discrete Applied Mathematics 157, 3174–3180 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biedl, T., Stern, M.: On edge intersection graphs of k-bend paths in grids. Discrete Mathematics & Theoretical Computer Science (DMTCS) 12, 1–12 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Cameron, K., Chaplick, S., Hoang, C.T.: Recognizing Edge Intersection Graphs of \(\llcorner\)-Shaped Grid Paths. In: LAGOS 2013 (to appear, 2013)

    Google Scholar 

  5. Cohen, E., Golumbic, M.C., Ries, B.: Characterizations of cographs as intersection graphs of paths on a grid (submitted)

    Google Scholar 

  6. Garey, M.R., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  7. Garey, M.R., Johnson, D.S., Miller, G.L., Papadimitriou, C.: The complexity of coloring circular arcs and chords. SIAM. J. on Algebraic and Discrete Methods 1, 216–227 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980); Annals of Discrete Mathematics, 2nd edn., vol. 57. Elsevier, Amsterdam (2004)

    Google Scholar 

  9. Golumbic, M.C., Lipshteyn, M., Stern, M.: Edge intersection graphs of single bend paths on a grid. Networks 54, 130–138 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Halldórsson, M.M.: A still better performance guarantee for approximate graph coloring. Information Processing Letters 45, 19–23 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heldt, D., Knauer, K., Ueckerdt, T.: Edge-intersection graphs of grid paths: the bend-number, Arxiv preprint arXiv:1009.2861, arxiv.org (September 2010)

    Google Scholar 

  12. Heldt, D., Knauer, K., Ueckerdt, T.: On the bend-number of planar and outerplanar graphs. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 458–469. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  13. Kako, A., Ono, T., Hirata, T., Halldórsson, M.M.: Approximation algorithms for the weighted independent set problem in sparse graphs. Discrete Applied Mathematics 157, 617–626 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kratochvíl, J., Nešetřil, J.: Independent set and clique problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae 31, 85–93 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Spinrad, J.P., Sritharan, R.: Algorithms for weakly triangulated graphs. Discrete Appl. Math. 59, 181–191 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Comput. 30, 135–140 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Haifa, Israel

    Dror Epstein & Martin Charles Golumbic

  2. Caesarea Rothschild Institute (CRI), University of Haifa, Israel

    Dror Epstein, Martin Charles Golumbic & Gila Morgenstern

Authors
  1. Dror Epstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Charles Golumbic
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gila Morgenstern
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computer Science, Parallel Computing and Bioinformatics Laboratory, Carleton University, VISM Building, Room 6210, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada

    Frank Dehne

  2. Department of Computer Science, The University of Western Ontario, N6A 5B7, London, ON, Canada

    Roberto Solis-Oba

  3. School of Computer Science, Carleton University, K1S 5B6, Ottawa, ON, Canada

    Jörg-Rüdiger Sack

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Epstein, D., Golumbic, M.C., Morgenstern, G. (2013). Approximation Algorithms for B 1-EPG Graphs. In: Dehne, F., Solis-Oba, R., Sack, JR. (eds) Algorithms and Data Structures. WADS 2013. Lecture Notes in Computer Science, vol 8037. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40104-6_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40104-6_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40103-9

  • Online ISBN: 978-3-642-40104-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation