Skip to main content
SpringerLink
Log in

Algorithmic Meta Theorems for Sparse Graph Classes

  • Conference paper
Computer Science - Theory and Applications (CSR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8476))

Included in the following conference series:

Abstract

Algorithmic meta theorems give efficient algorithms for classes of algorithmic problems, instead of just individual problems. They unify families of algorithmic results obtained by similar techniques and thus exhibit the core of these techniques. The classes of problems are typically defined in terms of logic and structural graph theory. A well-known example of an algorithmic meta theorem is Courcelle’s Theorem, stating that all properties of graphs of bounded tree width that are definable in monadic second-order logic are decidable in linear time.

This paper is a brief and nontechnical survey of the most important algorithmic meta theorems.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Bagan, G.: MSO queries on tree decomposable structures are computable with linear delay. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 167–181. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  3. Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 194–242. Elsevier Science Publishers (1990)

    Google Scholar 

  4. Courcelle, B.: Linear delay enumeration and monadic second-order logic. Discrete Applied Mathematics 157(12), 2675–2700 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Courcelle, B., Makowsky, J., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique width. Theory of Computing Systems 33(2), 125–150 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Courcelle, B., Makowsky, J., Rotics, U.: On the fixed-parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Applied Mathematics 108(1-2), 23–52 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science 109, 49–82 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: Proceedings of the 22nd IEEE Symposium on Logic in Computer Science, pp. 270–279 (2007)

    Google Scholar 

  9. Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Approximation schemes for first-order definable optimisation problems. In: Proceedings of the 21st IEEE Symposium on Logic in Computer Science, pp. 411–420 (2006)

    Google Scholar 

  10. Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer (2013)

    Google Scholar 

  11. Durand, A., Schweikardt, N., Segoufin, L.: Enumerating first-order queries over databases of low degree. In: Proceedings of the 33rd ACM Symposium on Principles of Database Systems (2014)

    Google Scholar 

  12. Durand, A., Grandjean, E.: First-order queries on structures of bounded degree are computable with constant delay. ACM Transactions on Computational Logic 8(4) (2007)

    Google Scholar 

  13. Dvořák, Z., Král, D., Thomas, R.: Deciding first-order properties for sparse graphs. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 133–142 (2010)

    Google Scholar 

  14. Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of bodlaender and courcelle. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 143–152 (2010)

    Google Scholar 

  15. Elberfeld, M., Jakoby, A., Tantau, T.: Algorithmic meta theorems for circuit classes of constant and logarithmic depth. In: Dürr, C., Wilke, T. (eds.) Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science. LIPIcs, vol. 14, pp. 66–77. Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik (2012)

    Google Scholar 

  16. Elberfeld, M., Kawarabayashi, K.I.: Embedding and canonizing graphs of bounded genus in logspace. In: Proceedings of the 46th ACM Symposium on Theory of Computing (2014)

    Google Scholar 

  17. Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. Journal of the ACM 49(6), 716–752 (2002)

    Article  MathSciNet  Google Scholar 

  18. Flum, J., Grohe, M.: Fixed-parameter tractability, definability, and model checking. SIAM Journal on Computing 31(1), 113–145 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)

    Google Scholar 

  20. Frick, M.: Generalized model-checking over locally tree-decomposable classes. Theory of Computing Systems 37(1), 157–191 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Annals of Pure and Applied Logic 130, 3–31 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ganian, R., Hliněný, P., Langer, A., Obdrlek, J., Rossmanith, P., Sikdar, S.: Lower bounds on the complexity of MSO1 model-checking. In: Dürr, C., Wilke, T. (eds.) Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science. LIPIcs, vol. 14, pp. 326–337. Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik (2012)

    Google Scholar 

  24. Ganian, R., Hliněný, P., Král’, D., Obdržálek, J., Schwartz, J., Teska, J.: FO model checking of interval graphs. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 250–262. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  25. Grohe, M.: Generalized model-checking problems for first-order logic. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 12–26. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  26. Grohe, M.: Logic, graphs, and algorithms. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata – History and Perspectives. Texts in Logic and Games, vol. 2, pp. 357–422. Amsterdam University Press (2007)

    Google Scholar 

  27. Grohe, M., Kawarabayashi, K., Reed, B.: A simple algorithm for the graph minor decomposition – logic meets structural graph theory. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 414–431 (2013)

    Google Scholar 

  28. Grohe, M., Kreutzer, S.: Methods for algorithmic meta theorems. In: Grohe, M., Makowsky, J. (eds.) Model Theoretic Methods in Finite Combinatorics. Contemporary Mathematics, vol. 558, pp. 181–206. American Mathematical Society (2011)

    Google Scholar 

  29. Grohe, M., Kreutzer, S., Siebertz, S.: Characterisations of nowhere denese graphs. In: Seth, A., Vishnoi, N. (eds.) Proceedings of the 32nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. LIPIcs, vol. 24, pp. 21–40. Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik (2013)

    Google Scholar 

  30. Grohe, M., Kreutzer, S., Siebertz, S.: Deciding first-order properties of nowhere dense graphs. In: Proceedings of the 46th ACM Symposium on Theory of Computing (2014)

    Google Scholar 

  31. Kazana, W., Segoufin, L.: First-order query evaluation on structures of bounded degree. Logical Methods in Computer Science 7(2) (2011)

    Google Scholar 

  32. Kazana, W., Segoufin, L.: Enumeration of first-order queries on classes of structures with bounded expansion. In: Proceedings of the 32nd ACM Symposium on Principles of Database Systems, pp. 297–308 (2013)

    Google Scholar 

  33. Kreutzer, S.: Algorithmic meta-theorems. In: Esparza, J., Michaux, C., Steinhorn, C. (eds.) Finite and Algorithmic Model Theory. London Mathematical Society Lecture Note Series, ch. 5, pp. 177–270. Cambridge University Press (2011)

    Google Scholar 

  34. Kreutzer, S., Tazari, S.: Lower bounds for the complexity of monadic second-order logic. In: Proceedings of the 25th IEEE Symposium on Logic in Computer Science, pp. 189–198 (2010)

    Google Scholar 

  35. Kreutzer, S., Tazari, S.: On brambles, grid-like minors, and parameterized intractability of monadic second-order logic. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 354–364 (2010)

    Google Scholar 

  36. Kreutzer, S.: On the parameterised intractability of monadic second-order logic. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 348–363. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  37. Langer, A., Reidl, F., Rossmanith, P., Sikdar, S.: Evaluation of an mso-solver. In: Proceedings of the 14th Meeting on Algorithm Engineering & Experiments, pp. 55–63 (2012)

    Google Scholar 

  38. Nešetřil, J., Ossona de Mendez, P.: On nowhere dense graphs. European Journal of Combinatorics 32(4), 600–617 (2011)

    MATH  MathSciNet  Google Scholar 

  39. Nešetřil, J., Ossona de Mendez, P.: Sparsity. Springer (2012)

    Google Scholar 

  40. Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  41. Seese, D.: Linear time computable problems and first-order descriptions. Mathematical Structures in Computer Science 6, 505–526 (1996)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RWTH Aachen University, Germany

    Martin Grohe

Authors
  1. Martin Grohe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, 27 Fontanka,, 191023, St. Petersburg, Russia

    Edward A. Hirsch

  2. Higher School of Economics, National Research University, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. CNRS and University Paris Diderot, Paris, France

    Jean-Éric Pin

  4. Moscow State University, 119992, Moscow, Russia

    Nikolay K. Vereshchagin

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Grohe, M. (2014). Algorithmic Meta Theorems for Sparse Graph Classes. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-06686-8_2

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06685-1

  • Online ISBN: 978-3-319-06686-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation