, Volume 70, Issue 1, pp 129–147 | Cite as

Towards Optimal and Expressive Kernelization for d-Hitting Set



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 (whose cardinality is bounded from above by a constant 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 \(\operatorname {coNP}\subseteq \operatorname {NP/poly}\)) and provide experimental results that show the practical applicability of our algorithm.

Finally, 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.


Parameterized algorithmics Linear-time data reduction Vertex cover in hypergraphs Fault diagnosis Sunflower lemma Algorithm engineering 



The author is very thankful to Rolf Niedermeier and the anonymous referees for many valuable comments.


  1. 1.
    Abtreu, R., Zoeteweij, P., van Gemund, A.J.C.: A dynamic modeling approach to software multiple-fault localization, Blue Mountains, NSW, Australia. In: Proc. 19th DX, pp. 7–14 (2008) Google Scholar
  2. 2.
    Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms. Addison-Wesley, Reading (1983) MATHGoogle Scholar
  4. 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: Proc. 6th IPEC. LNCS, vol. 7112, pp. 194–206. Springer, Berlin (2012) Google Scholar
  5. 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) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bodlaender, H.L.: Kernelization: new upper and lower bound techniques. In: Proc. 4th IWPEC. LNCS, vol. 5917, pp. 17–37. Springer, Berlin (2009) Google Scholar
  7. 7.
    Brankovic, L., Fernau, H.: Parameterized approximation algorithms for hitting set. In: Proc. 9th WAOA, pp. 63–76. Springer, Berlin (2012) Google Scholar
  8. 8.
    Damaschke, P.: Parameterized enumeration, transversals, and imperfect phylogeny reconstruction. Theor. Comput. Sci. 351(3), 337–350 (2006) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    de Kleer, J., Williams, B.C.: Diagnosing multiple faults. Artif. Intell. 32(1), 97–130 (1987) CrossRefMATHGoogle Scholar
  10. 10.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd STOC, pp. 251–260. ACM, New York (2010) Google Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  12. 12.
    Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 35, 85–90 (1960) CrossRefGoogle Scholar
  13. 13.
    Fernau, H.: Edge dominating set: efficient enumeration-based exact algorithms. In: Proc. 2nd IWPEC. LNCS, vol. 4169, pp. 142–153. Springer, Berlin (2006) Google Scholar
  14. 14.
    Fernau, H.: Parameterized algorithms for d-hitting set: the weighted case. Theor. Comput. Sci. 411(16–18), 1698–1713 (2010) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  16. 16.
    Fomin, F.V., Saurabh, S., Villanger, Y.: A polynomial kernel for proper interval vertex deletion. In: Proc. 2nd ESA. LNCS, vol. 7501, pp. 467–478. Springer, Berlin (2012) Google Scholar
  17. 17.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007) CrossRefGoogle Scholar
  18. 18.
    Hagerup, T.: Linear-time kernelization for planar dominating set. In: Proc. 6th IPEC. LNCS, vol. 7112, pp. 181–193. Springer, Berlin (2011) Google Scholar
  19. 19.
    Hagerup, T.: Kernels for edge dominating set: simpler or smaller. In: Proc. 37th MFCS. LNCS, vol. 7464, pp. 491–502. Springer, Berlin (2012) Google Scholar
  20. 20.
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47(1), 196–217 (2010) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2−ε. J. Comput. Syst. Sci. 74(3), 335–349 (2008) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Kratsch, S.: Polynomial kernelizations for \(\mbox{MIN F}^{+} \mathrm{\varPi}_{1}\) and MAX NP. Algorithmica 63(1), 532–550 (2012) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Moser, H.: Finding optimal solutions for covering and matching problems. PhD thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena (2010) Google Scholar
  24. 24.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, New York (2006) CrossRefMATHGoogle Scholar
  25. 25.
    Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algorithms 1(1), 89–102 (2003) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Nishimura, N., Ragde, P., Thilikos, D.M.: Smaller kernels for hitting set problems of constant arity. In: Proc. 1st IWPEC. LNCS, vol. 3162, pp. 121–126. Springer, Berlin (2004) Google Scholar
  27. 27.
    Protti, F., Dantas da Silva, M., Szwarcfiter, J.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. 44, 91–104 (2009) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. A. Springer, Berlin (2003) Google Scholar
  30. 30.
    Shi, L., Cai, X.: An exact fast algorithm for minimum hitting set. In: Proc. 3rd CSO, pp. 64–67. IEEE Computer Society, Los Alamitos (2010) Google Scholar
  31. 31.
    Sorge, M., Moser, H., Niedermeier, R., Weller, M.: Exploiting a hypergraph model for finding Golomb rulers. In: Proc. 2nd ISCO. LNCS, vol. 7422, pp. 368–379. Springer, Berlin (2012) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

Personalised recommendations