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European Symposium on Algorithms

ESA 2010: Algorithms – ESA 2010 pp 549–560Cite as

Algorithmic Meta-theorems for Restrictions of Treewidth

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

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.

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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 

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Author information

Authors and Affiliations

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

    Michael Lampis

Authors
  1. Michael Lampis
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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

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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

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  • 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)

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