Skip to main content
Springer Nature Link
Log in

Even Faster Algorithm for Set Splitting!

  • Conference paper
Parameterized and Exact Computation (IWPEC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5917))

Included in the following conference series:

  • 722 Accesses

Abstract

In p -Set Splitting we are given a universe U, a family \(\cal F\) of subsets of U and a positive integer k and the objective is to find a partition of U into W and B such that there are at least k sets in \(\cal F\) that have non-empty intersection with both B and W. In this paper we study p -Set Splitting from kernelization and algorithmic view points. Given an instance \((U,{\cal F},k)\) of p -Set Splitting, our kernelization algorithm obtains an equivalent instance with at most 2k sets and k elements in polynomial time. Finally, we give a fixed parameter tractable algorithm for p -Set Splitting running in time O(1.9630k + N), where N is the size of the instance. Both our kernel and our algorithm improve over the best previously known results. Our kernelization algorithm utilizes a classical duality theorem for a connectivity notion in hypergraphs. We believe that the duality theorem we make use of, will turn out to be an important tool from combinatorial optimization in obtaining kernelization algorithms.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Andersson, G., Engebretsen, L.: Better approximation algorithms for SET SPLITTING and NOT-ALL-EQUAL SAT. Inf. Process. Lett. 65(6), 305–311 (1998)

    Article  MathSciNet  Google Scholar 

  2. Chen, J., Lu, S.: Improved algorithms for weighted and unweighted set splitting problems. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 537–547. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  3. Chlebík, M., Chlebíková, J.: Crown reductions for the minimum weighted vertex cover problem. Discrete Applied Mathematics 156(3), 292–312 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dehne, F.K.H.A., Fellows, M.R., Rosamond, F.A.: An FPT algorithm for set splitting. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 180–191. Springer, Heidelberg (2003)

    Google Scholar 

  5. Dehne, F.K.H.A., Fellows, M.R., Rosamond, F.A., Shaw, P.: Greedy localization, iterative compression, modeled crown reductions: New fpt techniques, an improved algorithm for set splitting, and a novel 2k kernelization for vertex cover. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 271–280. Springer, Heidelberg (2004)

    Google Scholar 

  6. Erdős, P.: On a combinatorial problem, I. Nordisk Mat. Tidskrift 11, 5–10 (1963)

    Google Scholar 

  7. Erdős, P.: On a combinatorial problem, II. Acta Math, Hungary 15, 445–447 (1964)

    Article  Google Scholar 

  8. Fellows, M.R.: Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)

    Google Scholar 

  9. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: Domination - a case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)

    Google Scholar 

  10. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: a simple o(2\(^{\mbox{0.288{\it }}}\)) independent set algorithm. In: SODA, pp. 18–25 (2006)

    Google Scholar 

  11. Frank, A., Király, T., Kriesell, M.: On decomposing a hypergraph into k connected sub-hypergraphs. Discrete Applied Mathematics 131(2), 373–383 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)

    Google Scholar 

  14. Lokshtanov, D., Sloper, C.: Fixed parameter set splitting, linear kernel and improved running time. In: ACiD. Texts in Algorithmics, vol. 4, pp. 105–113 (2005)

    Google Scholar 

  15. Lovász, L.: Covering and coloring of hypergraphs. In: Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing. Utilitas Mathematica Publishing, pp. 3–12 (1973)

    Google Scholar 

  16. Prieto, E.: The method of extremal structure on the k-maximum cut problem. In: CATS, pp. 119–126 (2005)

    Google Scholar 

  17. Radhakrishnan, J., Srinivasan, A.: Improved bounds and algorithms for hypergraph 2-coloring. Random Struct. Algorithms 16(1), 4–32 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Raman, V., Saurabh, S.: Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG. Inf. Process. Lett. 104(2), 65–72 (2007)

    Article  MathSciNet  Google Scholar 

  19. Sloper, C.: Techniques in parameterized algorithm design. PhD thesis, University of Bergen (2005)

    Google Scholar 

  20. Wahlström, M.: Algorithms, measures, and upper bounds for satisfiability and related problems. PhD thesis, Linkp̈ing University (2007)

    Google Scholar 

  21. Zhang, J., Ye, Y., Han, Q.: Improved approximations for max set splitting and max NAE SAT. Discrete Applied Mathematics 142(1-3), 133–149 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zwick, U.: Outward rotations: A tool for rounding solutions of semidefinite programming relaxations, with applications to max cut and other problems. In: STOC, pp. 679–687 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    Daniel Lokshtanov

  2. The Institute of Mathematical Sciences, C.I.T. Campus, Chennai, 600 113

    Saket Saurabh

Authors
  1. Daniel Lokshtanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saket Saurabh
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Texas A&M University, College Station, 77843, Texas, USA

    Jianer Chen

  2. Institutt for informatikk, Universitetet i Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lokshtanov, D., Saurabh, S. (2009). Even Faster Algorithm for Set Splitting!. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-11269-0_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11268-3

  • Online ISBN: 978-3-642-11269-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics