iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://doi.org/10.1007/978-3-642-15775-2_47
Algorithmic Meta-theorems for Restrictions of Treewidth | SpringerLink
Skip to main content
Springer Nature Link
Log in

Algorithmic Meta-theorems for Restrictions of Treewidth

  • Conference paper
Algorithms – ESA 2010 (ESA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6346))

Included in the following conference series:

Abstract

Possibly the most famous algorithmic meta-theorem is Courcelle’s theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time’s dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees.

We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number. We prove stronger algorithmic meta-theorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.

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. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, J., Kanj, I.A., Xia, G.: Improved parameterized upper bounds for vertex cover. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 238–249. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  3. Courcelle, B.: The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Approximation schemes for first-order definable optimisation problems. In: LICS, pp. 411–420. IEEE Computer Society, Los Alamitos (2006)

    Google Scholar 

  6. Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-Time Extremal Structure I. In: Broersma, H., Johnson, M., Szeider, S. (eds.) ACiD. Texts in Algorithmics, vol. 4, pp. 1–41. King’s College, London (2005)

    Google Scholar 

  7. Fellows, M.R.: Open problems in parameterized complexity. In: AGAPE spring school on fixed parameter and exact algorithms (2009)

    Google Scholar 

  8. Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number. Theory Comput. Syst. 45(4), 822–848 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 294–305. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  10. Fellows, M.R., Rosamond, F.A.: The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, pp. 268–277. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  11. Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Clique-width: on the price of generality. In: Mathieu, C. (ed.) SODA, pp. 825–834. SIAM, Philadelphia (2009)

    Google Scholar 

  12. Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. J. ACM 48(6), 1184–1206 (2001)

    Article  MathSciNet  Google Scholar 

  13. Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130(1-3), 3–31 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grohe, M.: Logic, graphs, and algorithms. Electronic Colloquium on Computational Complexity (ECCC) 14(091) (2007)

    Google Scholar 

  15. Hlinený, P., il Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. Comput. J. 51(3), 326–362 (2008)

    Article  Google Scholar 

  16. Kleitman, D., West, D.: Spanning trees with many leaves. SIAM Journal on Discrete Mathematics 4, 99 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kreutzer, S., Tazari, S.: On brambles, grid-like minors, and parameterized intractability of monadic second order logic. In: SODA (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Graduate Center, City University of New York,  

    Michael Lampis

Authors
  1. Michael Lampis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands

    Mark de Berg

  2. Institute for Computer Science, J.W. Goethe University, 60325, Frankfurt/Main, Germany

    Ulrich Meyer

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lampis, M. (2010). Algorithmic Meta-theorems for Restrictions of Treewidth. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_47

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15775-2_47

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15774-5

  • Online ISBN: 978-3-642-15775-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation