Incompressibility through Colors and IDs

  • Michael Dom
  • Daniel Lokshtanov
  • Saket Saurabh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5555)


In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existence of polynomial kernels has been developed by Bodlaender et al. [4] and Fortnow and Santhanam [9]. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems:

  • We show that the Steiner Tree problem parameterized by the number of terminals and solution size k, and the Connected Vertex Cover and Capacitated Vertex Cover problems do not admit a polynomial kernel. The two latter results are surprising because the closely related Vertex Cover problem admits a kernel of size 2k.

  • Alon and Gutner obtain a k poly(h) kernel for Dominating Set in H -Minor Free Graphs parameterized by h = |H| and solution size k and ask whether kernels of smaller size exist [2]. We partially resolve this question by showing that Dominating Set in H -Minor Free Graphs does not admit a kernel with size polynomial in k + h.

  • Harnik and Naor obtain a “compression algorithm” for the Sparse Subset Sum problem [13]. We show that their algorithm is essentially optimal since the instances cannot be compressed further.

  • Hitting Set and Set Cover admit a kernel of size k O(d) when parameterized by solution size k and maximum set size d. We show that neither of them, along with the Unique Coverage and Bounded Rank Disjoint Sets problems, admits a polynomial kernel.

All results are under the assumption that the polynomial hierarchy does not collapse to the third level. The existence of polynomial kernels for several of the problems mentioned above were open problems explicitly stated in the literature [2,3,11,12,14]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [13], for the problems in question.


Vertex Cover Parameterized Problem Colored Version Polynomial Kernel Steiner Tree Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.
    Alon, N., Gutner, S.: Kernels for the dominating set problem on graphs with an excluded minor. Technical Report TR08-066, Electronic Colloquium on Computational Complexity (ECCC) (2008)Google Scholar
  3. 3.
    Betzler, N.: Steiner tree problems in the analysis of biological networks. Diploma thesis, Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany (2006)Google 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.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels. Technical Report UU-CS-2008-030, Institute of Information and Computing Sciences, Utrecht University, Netherlands (2008)Google Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fernau, H., Fomin, F.V., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: On out-trees with many leaves. In: Proc. 26th STACS (to appear)Google Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  9. 9.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proc. 40th STOC, pp. 133–142. ACM Press, New York (2008)Google Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  11. 11.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  12. 12.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of Vertex Cover variants. Theory Comput. Syst. 41(3), 501–520 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Harnik, D., Naor, M.: On the compressibility of NP instances and cryptographic applications. In: Proc. 47th FOCS, pp. 719–728. IEEE, Los Alamitos (2007)Google Scholar
  14. 14.
    Moser, H., Raman, V., Sikdar, S.: The parameterized complexity of the unique coverage problem. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 621–631. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Michael Dom
    • 1
  • Daniel Lokshtanov
    • 2
  • Saket Saurabh
    • 2
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.Department of InformaticsUniversity of BergenBergenNorway

Personalised recommendations