Advertisement

Kernelization Algorithms for d-Hitting Set Problems

  • Faisal N. Abu-Khzam
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4619)

Abstract

A kernelization algorithm for the 3-Hitting-Set problem is presented along with a general kernelization for d-Hitting-Set problems. For 3-Hitting-Set, a quadratic kernel is obtained by exploring properties of yes instances and employing what is known as crown reduction. Any 3-Hitting-Set instance is reduced into an equivalent instance that contains at most 5k 2 + k elements (or vertices). This kernelization is an improvement over previously known methods that guarantee cubic-size kernels. Our method is used also to obtain a quadratic kernel for the Triangle Vertex Deletion problem. For a constant d ≥ 3, a kernelization of d-Hitting-Set is achieved by a generalization of the 3-Hitting-Set method, and guarantees a kernel whose order does not exceed (2d − 1)k d − 1 + k.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization Algorithms for the Vertex Cover Problem: Theory and Experiments. In: ALENEX. Workshop on Algorithm Engineering and Experiments, pp. 62–69 (2004)Google Scholar
  2. 2.
    Abu-Khzam, F.N., Fellows, M.R., Langston, M.A., Suters, W.H.: Crown Structures for Vertex Cover Kernelization. Theory of Computing Systems (TOCS) (accepted for publication)Google Scholar
  3. 3.
    Abu-Khzam, F.N., Fernau, H.: Kernels: Annotated, Proper and Induced. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 264–275. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Buss, J.F., Goldsmith, J.: Nondeterminism within P. SIAM Journal on Computing 22, 560–572 (1993)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, J., Kanj, I., Jia, W.: Vertex cover: further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chor, B., Fellows, M.R., Juedes, D.: Linear Kernels in Linear Time, or How to Save k Colors in O(n 2) Steps. In: International Workshop on Graph Theoretic Concepts in Computer Science (WG), pp. 257–269 (2004)Google Scholar
  7. 7.
    Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-Hitting Set. Electronic Colloquium on Computational Complexity (ECCC), p. 073 (2004)Google Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman, New York (1979)MATHGoogle Scholar
  9. 9.
    Jones, J.A., Harrold, M.J.: Test-Suite Reduction and Prioritization for Modified Condition/Decision Coverage. IEEE Trans. Software Eng. 29(3), 195–209 (2003)CrossRefGoogle Scholar
  10. 10.
    Kuhn, F., von Rickenbach, P., Wattenhofer, R., Welzl, E., Zollinger, A.: Interference in Cellular Networks: The Minimum Membership Set Cover Problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Nemhauser, G.L., Trotter, L.E.: Vertex Packings: Structural Properties and Algorithms. Mathematical Programming 8, 232–248 (1975)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-Hitting Set. Journal of Discrete Algorithms 1, 89–102 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nishimura, N., Ragde, P., Thilikos, D.M.: Smaller kernels for hitting set problems of constant arity. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 121–126. Springer, Heidelberg (2004)Google Scholar
  14. 14.
    Ruchkys, D., Song, S.: A Parallel Approximation Hitting Set Algorithm for Gene Expression Analysis. In: SBAC-PAD 2002. Proceedings of the 14th Symposium on Computer Architecture and High Performance Computing, Washington, DC, USA, p. 75. IEEE Computer Society, Los Alamitos (2002)Google Scholar
  15. 15.
    Thilikos, D.M.: Private communication (2005)Google Scholar
  16. 16.
    Weihe, K.: Covering trains by stations or the power of data reduction. In: Battiti, R., Bertossi, A.A. (eds.) International Conference on Algorithms and Experiments, pp. 1–8 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Faisal N. Abu-Khzam
    • 1
  1. 1.Division of Computer Science and Mathematics, Lebanese American University, BeirutLebanon

Personalised recommendations