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Towards Optimal and Expressive Kernelization for d-Hitting Set

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Computing and Combinatorics (COCOON 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7434))

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Abstract

A sunflower in a hypergraph is a set of hyperedges pairwise intersecting in exactly the same vertex set. Sunflowers are a useful tool in polynomial-time data reduction for problems formalizable as d-Hitting Set, the problem of covering all hyperedges (of cardinality at most d) of a hypergraph by at most k vertices. Additionally, in fault diagnosis, sunflowers yield concise explanations for “highly defective structures”. We provide a linear-time algorithm that, by finding sunflowers, transforms an instance of d-Hitting Set into an equivalent instance comprising at most O(k d) hyperedges and vertices. In terms of parameterized complexity, we show a problem kernel with asymptotically optimal size (unless coNP ⊆ NP/poly). We show that the number of vertices can be reduced to O(k d − 1) with additional processing in O(k 1.5d) time—nontrivially combining the sunflower technique with problem kernels due to Abu-Khzam and Moser.

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References

  1. Abtreu, R., Zoeteweij, P., van Gemund, A.J.C.: A dynamic modeling approach to software multiple-fault localization. In: Proc. 19th DX, pp. 7–14. Blue Mountains, NSW, Australia (2008)

    Google Scholar 

  2. Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms. Addison-Wesley (1983)

    Google Scholar 

  4. van Bevern, R., Hartung, S., Kammer, F., Niedermeier, R., Weller, M.: Linear-Time Computation of a Linear Problem Kernel for Dominating Set on Planar Graphs. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 194–206. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  5. van Bevern, R., Moser, H., Niedermeier, R.: Approximation and tidying—a problem kernel for s-plex cluster vertex deletion. Algorithmica 62(3), 930–950 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  7. Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd STOC 2010, pp. 251–260. ACM (2010)

    Google Scholar 

  8. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)

    Google Scholar 

  9. Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. London Math. Soc. 35, 85–90 (1960)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  11. Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)

    Article  Google Scholar 

  12. Hagerup, T.: Linear-Time Kernelization for Planar Dominating Set. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 181–193. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  13. Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47(1), 196–217 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Kleer, J., Williams, B.C.: Diagnosing multiple faults. Artif. Intell. 32(1), 97–130 (1987)

    Article  MATH  Google Scholar 

  15. Kratsch, S.: Polynomial kernelizations for MIN F +  Π1 and MAX NP. Algorithmica 63(1), 532–550 (2012), ISSN 0178-4617

    Article  MathSciNet  MATH  Google Scholar 

  16. Moser, H.: Finding Optimal Solutions for Covering and Matching Problems. PhD thesis. Institut für Informatik, Friedrich-Schiller-Universität Jena (2010)

    Google Scholar 

  17. Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, USA (2006)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Chapter  Google Scholar 

  20. Protti, F., Dantas da Silva, M., Szwarcfiter, J.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. 44, 91–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. A. Springer (2003)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany

    René van Bevern

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  1. René van Bevern
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Editor information

Editors and Affiliations

  1. School of IT, University of Sydney, Building J12, 2006, Sydney, NSW, Australia

    Joachim Gudmundsson , Julián Mestre  & Taso Viglas ,  & 

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van Bevern, R. (2012). Towards Optimal and Expressive Kernelization for d-Hitting Set. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_11

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  • DOI: https://doi.org/10.1007/978-3-642-32241-9_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32240-2

  • Online ISBN: 978-3-642-32241-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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