Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

How to Use Planarity Efficiently: New Tree-Decomposition Based Algorithms

  • Conference paper
Graph-Theoretic Concepts in Computer Science (WG 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4769))

Included in the following conference series:

  • 809 Accesses

Abstract

We prove new structural properties for tree-decompositions of planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. We give for example the fastest algorithm for Planar Dominating Set of runtime 3tw ·n O(1), when we take the treewidth tw as the measure for the exponential worst case behavior. We also introduce a tree-decomposition based approach to solve non-local problems efficiently, such as Planar Hamiltonian Cycle in runtime 6tw ·n O(1). From any input tree-decomposition, we compute in time O(nm) a tree-decomposition with geometric properties, which decomposes the plane into disks, and where the graph separators form Jordan curves in the plane.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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.

Similar content being viewed by others

References

  1. Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alber, J., Bodlaender, H.L., Fernau, H., Niedermeier, R.: Fixed parameter algorithms for planar dominating set and related problems. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 97–110. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  3. Alber, J., Niedermeier, R.: Improved tree decomposition based algorithms for domination-like problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–627. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  4. Arora, S., Grigni, M., Karger, D., Klein, P., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, pp. 33–41. ACM Press, New York (1998)

    Google Scholar 

  5. Bernstein, P.A., Goodman, N.: Power of natural semijoins. SIAM Journal on Computing 10, 751–771 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bodlaender, H.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  7. Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernet. 11, 1–21 (1993)

    MATH  MathSciNet  Google Scholar 

  8. Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM Journal on Discrete Mathematics 6, 181–188 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bouchitté, V., Mazoit, F., Todinca, I.: Chordal embeddings of planar graphs. Discrete Mathematics 273, 85–102 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cook, W., Seymour, P.: Tour merging via branch-decomposition. INFORMS Journal on Computing 15, 233–248 (2003)

    Article  MathSciNet  Google Scholar 

  11. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. Journal of the ACM 52, 866–893 (2005)

    Article  MathSciNet  Google Scholar 

  12. Diestel, R.: Graph theory, 3rd edn. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  13. Dorn, F.: Dynamic programming and fast matrix multiplication. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 280–291. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  14. Dorn, F.: How to use planarity efficiently: new tree-decomposition based algorithms (manuscript, 2007), http://www.ii.uib.no/~frederic/PlanDomSet.pdf

  15. Dorn, F., Fomin, F.V., Thilikos, D.M.: Fast subexponential algorithm for non-local problems on graphs of bounded genus. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 172–183. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  16. Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 95–106. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  17. Dorn, F., Telle, J.A.: Two birds with one stone: the best of branchwidth and treewidth with one algorithm. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 386–397. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  18. Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3, 1–27 (1999)

    Article  MathSciNet  Google Scholar 

  19. Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: branch-width and exponential speed-up. In: SODA 2003. Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, pp. 168–177. ACM Press, New York (2003)

    Google Scholar 

  20. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory Series B 16, 47–56 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  21. Heggernes, P.: Minimal triangulations of graphs: A survey. Discrete Mathematics 306, 297–317 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ho, C.W., Lee, R.C.T.: Counting clique trees and computing perfect elimination schemes in parallel. Inf. Process. Lett. 31, 61–68 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  24. Miller, G.L.: Finding small simple cycle separators for 2-connected planar graphs. Journal of Computer and System Science 32, 265–279 (1986)

    Article  MATH  Google Scholar 

  25. Parra, A., Scheffler, P.: Characterizations and algorithmic applications of chordal graph embeddings. Discrete Applied Mathematics 79, 171–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Parter, S.: The use of linear graphs in Gauss elimination. SIAM Review 3, 119–130 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  27. Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52, 153–190 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rose, D., Tarjan, R.E., Lueker, G.: Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing 5, 146–160 (1976)

    Article  MathSciNet  Google Scholar 

  29. Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing 13, 566–579 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  30. Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math 10, 529–550 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. van Leeuwen, J.: Graph algorithms. MIT Press, Cambridge, MA, USA (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, PO Box 7800, 5020 Bergen, Norway

    Frederic Dorn

Authors
  1. Frederic Dorn
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Andreas Brandstädt Dieter Kratsch Haiko Müller

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dorn, F. (2007). How to Use Planarity Efficiently: New Tree-Decomposition Based Algorithms. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-74839-7_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74838-0

  • Online ISBN: 978-3-540-74839-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation