Skip to main content
SpringerLink
Log in

Fast Algorithms for max independent set

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We first propose a method, called “bottom-up method” that, informally, “propagates” improvement of the worst-case complexity for “sparse” instances to “denser” ones and we show an easy though non-trivial application of it to the min set cover problem. We then tackle max independent set. Here, we propagate improvements of worst-case complexity from graphs of average degree d to graphs of average degree greater than d. Indeed, using algorithms for max independent set in graphs of average degree 3, we successively solve max independent set in graphs of average degree 4, 5 and 6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree 5 and 6 but also for general graphs. The computation bounds obtained for max independent set are O ∗(1.1571n), O ∗(1.1895n) and O ∗(1.2050n), for graphs of maximum (or more generally average) degree 4, 5 and 6 respectively, and O ∗(1.2114n) for general graphs. These results improve upon the best known results for these cases for polynomial space algorithms.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beigel, R.: Finding maximum independent sets in sparse and general graphs. In: Proc. Symposium on Discrete Algorithms, SODA’99, pp. 856–857 (1999)

    Google Scholar 

  2. Bourgeois, N., Escoffier, B., Paschos, V.Th.: An O ∗(1.0977n) exact algorithm for max independent set in sparse graphs. In: Grohe, M., Niedermeier, R. (eds.) Proc. International Workshop on Exact and Parameterized Computation, IWPEC’08. Lecture Notes in Computer Science, vol. 5018, pp. 55–65. Springer, Berlin (2008)

    Chapter  Google Scholar 

  3. Bourgeois, N., Escoffier, B., Paschos, V.Th., Van Rooij, J.M.M: Maximum independent set in graphs of average degree at most three in O(1.08537n). In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) Proc. Conference on Theory and Applications of Models of Computation, TAMC’10. Lecture Notes in Computer Science, vol. 6108, pp. 373–384. Springer, Berlin (2010)

    Chapter  Google Scholar 

  4. Chen, J., Liu, L., Jia, W.: Improvement on vertex cover for low-degree graphs. Networks 35(4), 253–259 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, J., Kanj, I.A., Xia, G.: Labeled search trees and amortized analysis: improved upper bounds for NP-hard problems. Algorithmica 43(4), 245–273 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2SAT and 3SAT formulæ. Theor. Comput. Sci. 332(1–3), 265–291 (2005)

    Article  MATH  Google Scholar 

  7. Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms. Inf. Process. Lett. 97(5), 191–196 (2006)

    Article  MATH  Google Scholar 

  8. Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. Assoc. Comput. Mach. 56(5), 1–32 (2009)

    Article  MathSciNet  Google Scholar 

  10. Fürer, M.: A faster algorithm for finding maximum independent sets in sparse graphs. In: Corea, J.R., Hevia, A., Kiwi, M. (eds.) Proc. Latin American Symposium on Theoretical Informatics, LATIN’06. Lecture Notes in Computer Science, vol. 3887, pp. 491–501. Springer, Berlin (2006)

    Chapter  Google Scholar 

  11. Fürer, M., Kasiviswanathan, S.P.: Algorithms for counting 2-SAT solutions and colorings with applications. In: Kao, M.-Y., Li, X.-Y. (eds.) Proc. Algorithmic Aspects in Information and Management, AAIM’07. Lecture Notes in Computer Science, vol. 4508, pp. 47–57. Springer, Berlin (2007)

    Chapter  Google Scholar 

  12. Goldberg, M.K., Spencer, T.H., Berque, D.A.: A low-exponential algorithm for counting vertex covers. In: Graph Theory, Combinatorics and Algorithms. Proc. 7th Quadrennial International Conference on the Theory and Application of Graphs, vol. 1, pp. 431–444. Wiley, New York (1992)

    Google Scholar 

  13. Kneis, J., Langer, A., Rossmanith, P.: A fine-grained analysis of a simple independent set algorithm. In: Kannan, R., Narayan Kumar, K. (eds.) Proc. Foundations of Software Technology and Theoretical Computer Science, FSTTCS’09. LIPIcs, vol. 4, pp. 287–298. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik (2009)

  14. Kojevnikov, A., Kulikov, A.S.: A new approach to proving upper bounds for max-2-sat. In: Proc. Symposium on Discrete Algorithms, SODA’06, pp. 11–17 (2006)

    Chapter  Google Scholar 

  15. Razgon, I.: Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3. J. Discrete Algorithms 7, 191–212 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7(3), 425–440 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  17. Robson, J.M.: Finding a maximum independent set in time O(2n/4). Technical Report 1251-01, LaBRI, Université de Bordeaux I (2001)

  18. van Rooij, J.M.M., Bodlaender, H.L.: Design by measure and conquer, a faster exact algorithm for dominating set. In: Albers, S., Weil, P. (eds.) Proc. International Symposium on Theoretical Aspects of Computer Science, STACS’08, pp. 657–668. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2008)

  19. Wahlström, M.: A tighter bound for counting max-weight solutions to 2sat instances. In: Grohe, M., Niedermeier, R. (eds.) Proc. Parameterized and Exact Computation, Third International Workshop, IWPEC’08. Lecture Notes in Computer Science, vol. 5018, pp. 202–213. Springer, Berlin (2008)

    Chapter  Google Scholar 

  20. Woeginger, G.J.: Exact algorithms for NP-hard problems: a survey. In: Juenger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization—Eureka! You shrink! Lecture Notes in Computer Science, vol. 2570, pp. 185–207. Springer, Berlin (2003)

    Chapter  Google Scholar 

  21. Xiao, M.: New branching rules: improvements on independent set and vertex cover in sparse graphs. CoRR (2009). arXiv:0904.2712

Download references

Author information

Authors and Affiliations

  1. LAMSADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cédex 16, France

    Nicolas Bourgeois, Bruno Escoffier & Vangelis T. Paschos

  2. Department of Information and Computing Sciences, Universiteit Utrecht, Utrecht, The Netherlands

    Johan M. M. van Rooij

Authors
  1. Nicolas Bourgeois
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bruno Escoffier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vangelis T. Paschos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Johan M. M. van Rooij
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno Escoffier.

Additional information

Research supported by the French Agency for Research under the DEFIS program TODO, ANR-09-EMER-010. Parts of this paper have been presented at SWAT’10, LNCS 6139 and at TAMC’10, LNCS 6108.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bourgeois, N., Escoffier, B., Paschos, V.T. et al. Fast Algorithms for max independent set . Algorithmica 62, 382–415 (2012). https://doi.org/10.1007/s00453-010-9460-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-010-9460-7

Keywords

Search

Navigation