Skip to main content
Log in

A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

In this paper, we show how to systematically improve on parameterized algorithms and their analysis, focusing on search-tree based algorithms for 3-Hitting Set. We concentrate on algorithms which are easy to implement, in contrast with the highly sophisticated algorithms which have been designed previously to improve on the exponential base in the algorithms.

However, this necessitates a more complex algorithm analysis based on a so-called auxiliary parameter, a technique which we believe can be used in other circumstances, as well.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abu-Khzam, F., Fernau, H.: Kernels: annotated, proper and induced. In: International Workshop on Parameterized and Exact Computation IWPEC. LNCS, vol. 4169, pp. 264–275. Springer, Berlin (2006)

    Chapter  Google Scholar 

  2. Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. In: Workshop on Algorithms and Data Structures WADS. LNCS, vol. 4619, pp. 434–445. Springer, Berlin (2007)

    Chapter  Google Scholar 

  3. Alber, J., Gramm, J., Niedermeier, R.: Faster exact algorithms for hard problems: a parameterized point of view. Discrete Math. 229(1), 3–27 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alber, J., Fan, H., Fellows, M.R., Fernau, H., Niedermeier, R., Rosamond, F., Stege, U.: A refined search tree techniques for dominating set on planar graphs. J. Comput. Syst. Sci. 71, 385–405 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anand, R.S., Erlebach, T., Hall, A., Stefanakos, S.: Call control with k rejections. J. Comput. Syst. Sci. 67, 707–722 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berry, V., Nicolas, F.: Maximum agreement and compatible supertrees. In: Combinatorial Pattern Matching Symposium (CPM). LNCS, vol. 3109, pp. 205–219. Springer, Berlin (2004)

    Google Scholar 

  7. Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41, 280–301 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, J., Kanj, I.A., Xia, G.: Improved parameterized upper bounds for vertex cover. In: Mathematical Foundations of Computer Science MFCS. LNCS, vol. 4162, pp. 238–249. Springer, Berlin (2006)

    Chapter  Google Scholar 

  10. Damaschke, P.: The union of minimal hitting sets: parameterized combinatorial bounds and counting. In: 24th Symposium on Theoretical Aspects of Computer Science STACS. LNCS, vol. 4393, pp. 332–343. Springer, Berlin (2007)

    Google Scholar 

  11. Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered PCP and the hardness of hypergraph vertex cover. In: Proc. 35th ACM Symp. on Theory of Computing (STOC), pp. 595–601 (2003)

  12. Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-parameter tractability results for feedback set problems in tournaments. In: Conference on Algorithms and Complexity CIAC. LNCS, vol. 3998, pp. 320–331. Springer, Berlin (2006)

    Google Scholar 

  13. Downey, R.G., Fellows, M.R., Stege, U.: Parameterized complexity: a framework for systematically confronting computational intractability. In: Contemporary Trends in Discrete Mathematics: From DIMACS and DIMATIA to the Future. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 49, pp. 49–99. AMS, Providence (1999)

    Google Scholar 

  14. Dujmović, V., Fellows, M.R., Hallett, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F.A., Suderman, M., Whitesides, S., Wood, D.R.: A fixed-parameter approach to two-layer planarization. In: 9th International Symposium on Graph Drawing GD. LNCS, vol. 2265, pp. 1–15. Springer, Berlin (2002)

    Google Scholar 

  15. Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-Hitting Set. Technical Report TR04-073, Electronic Colloquium on Computational Complexity ECCC (2004)

  16. Fernau, H.: Two-layer planarization: improving on parameterized algorithmics. J. Graph Algorithms Appl. 9, 205–238 (2005)

    MATH  MathSciNet  Google Scholar 

  17. Fernau, H.: Parameterized algorithms for hitting set: the weighted case. In: Conference on Algorithms and Complexity CIAC. LNCS, vol. 3998, pp. 332–343. Springer, Berlin (2006)

    Google Scholar 

  18. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: domination—a case study. In: Automata, Languages and Programming, 32nd International Colloquium, ICALP. LNCS, vol. 3580, pp. 191–203. Springer, Berlin (2005)

    Google Scholar 

  19. Garfinkel, R.S., Nemhauser, G.L.: Integer Programming. Wiley, New York (1972)

    MATH  Google Scholar 

  20. Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39, 321–347 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theor. Comput. Sci. 223, 1–72 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-Hitting Set. J. Discrete Algorithms 1, 89–102 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32, 57–95 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  24. Suderman, M.: Layered graph drawing. Ph.D. thesis, McGill University, Montréal (2005)

    Google Scholar 

  25. Wahlström, M.: Exact algorithms for finding minimum transversals in rank-3 hypergraphs. J. Algorithms 51, 107–121 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. thesis, Department of Computer and Information Science, Linköpings universitet, Sweden (2007). Available through http://www.diva-portal.org/diva/getDocument?urn_nbn_se_liu_diva-8714-1_fulltext.pdf

  27. Weihe, K.: Covering trains by stations or the power of data reduction. In: Algorithms and Experiments ALEX 98, pp. 1–8 (1998). http://rtm.science.unitn.it/alex98/proceedings.html

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Henning Fernau.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fernau, H. A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set . Algorithmica 57, 97–118 (2010). https://doi.org/10.1007/s00453-008-9199-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-008-9199-6

Keywords

Navigation