Preprocessing of Min Ones Problems: A Dichotomy

  • Stefan Kratsch
  • Magnus Wahlström
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

Min Ones Constraint Satisfaction Problems, i.e., the task of finding a satisfying assignment with at most k true variables (Min Ones SAT(Γ)), can express a number of interesting and natural problems. We study the preprocessing properties of this class of problems with respect to k, using the notion of kernelization to capture the viability of preprocessing. We give a dichotomy of Min Ones SAT(Γ) problems into admitting or not admitting a kernelization with size guarantee polynomial in k, based on the constraint language Γ. We introduce a property of boolean relations called mergeability that can be easily checked for any Γ. When all relations in Γ are mergeable, then we show a polynomial kernelization for Min Ones SAT(Γ). Otherwise, any Γ containing a non-mergeable relation and such that Min Ones SAT(Γ) is NP-complete permits us to prove that Min Ones SAT(Γ) does not admit a polynomial kernelization unless NP ⊆ co-NP/poly, by a reduction from a particular parameterization of Exact Hitting Set.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 434–445. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A cubic kernel for feedback vertex set. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 320–331. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels (extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 635–646. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Burrage, K., Estivill-Castro, V., Fellows, M.R., Langston, M.A., Mac, S., Rosamond, F.A.: The undirected feedback vertex set problem has a poly(k) kernel. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 192–202. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Creignou, N., Vollmer, H.: Boolean constraint satisfaction problems: When does Post’s lattice help? In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.) Complexity of Constraints. LNCS, vol. 5250, pp. 3–37. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: STOC (to appear, 2010)Google Scholar
  9. 9.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and ids. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity (Monographs in Computer Science). Springer, Heidelberg (November 1998)Google Scholar
  11. 11.
    Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. London Math. Soc. 35, 85–90 (1960)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: STOC, pp. 133–142. ACM, New York (2008)Google Scholar
  13. 13.
    Harnik, D., Naor, M.: On the compressibility of NP instances and cryptographic applications. SIAM J. Comput. 39(5), 1667–1713 (2010)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2000)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kratsch, S.: Polynomial kernelizations for MIN F\(^+{\Pi}_1\) and MAX NP. In: STACS 2009. Dagstuhl Seminar Proceedings, vol. 09001, pp. 601–612. Schloss Dagstuhl, Germany (2009)Google Scholar
  16. 16.
    Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. In: Chen, J., Fomin, F.V. (eds.) IWPEC. LNCS, vol. 5917, pp. 264–275. Springer, Heidelberg (2009)Google Scholar
  17. 17.
    Kratsch, S., Wahlström, M.: Preprocessing of min ones problems: a dichotomy. CoRR, abs/0910.4518 (2009)Google Scholar
  18. 18.
    Marx, D.: Parameterized complexity of constraint satisfaction problems. Computational Complexity 14(2), 153–183 (2005)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Moser, H.: A problem kernelization for graph packing. In: Nielsen, M., Kucera, A., Miltersen, P.B., Palamidessi, C., Tuma, P., Valencia, F.D. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 401–412. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Schaefer, T.J.: The complexity of satisfiability problems. In: STOC, pp. 216–226. ACM, New York (1978)Google Scholar
  21. 21.
    Thomassé, S.: A quadratic kernel for feedback vertex set. In: SODA, pp. 115–119. SIAM, Philadelphia (2009)Google Scholar
  22. 22.
    Yap, C.-K.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287–300 (1983)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Stefan Kratsch
    • 1
  • Magnus Wahlström
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations