Diminishable Parameterized Problems and Strict Polynomial Kernelization

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10936)


Kernelization—a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems—plays a central role in parameterized complexity and has triggered an extensive line of research. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and Müller [CiE 2009, Theory Comput. Syst. 2011], we underline the applicability of their framework by showing that a variety of fixed-parameter tractable problems, including graph problems and Turing machine computation problems, does not admit strict polynomial kernels under the weaker assumption of \( P {}\ne NP {}\). Finally, we study a relaxation of the notion of strict kernels.


  1. 1.
    Abu-Khzam, F.N., Fernau, H.: Kernels: annotated, proper and induced. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 264–275. Springer, Heidelberg (2006). Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Betzler, N., Guo, J., Komusiewicz, C., Niedermeier, R.: Average parameterization and partial kernelization for computing medians. J. Comput. Syst. Sci. 77(4), 774–789 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Binkele-Raible, D., Fernau, H., Fomin, F.V., Lokshtanov, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: on out-trees with many leaves. ACM Trans. Algorithms 8(4), 38 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: Advice classes of parameterized tractability. Ann. Pure Appl. Logic 84(1), 119–138 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: lower bounds and upper bounds on kernel size. SIAM J. Comput. 37(4), 1077–1106 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, Y., Flum, J., Müller, M.: Lower bounds for kernelizations and other preprocessing procedures. Theory Comput. Syst. 48(4), 803–839 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diestel, R.: Graph Theory, 4th edn. Springer, Heidelberg (2010). Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Heidelberg (2013). Scholar
  11. 11.
    Fellows, M.R., Kulik, A., Rosamond, F.A., Shachnai, H.: Parameterized approximation via fidelity preserving transformations. J. Comput. Syst. Sci. 93, 30–40 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fernau, H., Fluschnik, T., Hermelin, D., Krebs, A., Molter, H., Niedermeier, R.: Diminishable parameterized problems and strict polynomial kernelization. CoRR abs/1611.03739 (2018).
  13. 13.
    Fluschnik, T., Mertzios, G.B., Nichterlein, A.: Kernelization lower bounds for finding constant size subgraphs. CoRR abs/1710.07601 (2017).
  14. 14.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  16. 16.
    Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Karp, R.M., Lipton, R.: Turing machines that take advice. L’Enseignement Mathématique 28(2), 191–209 (1982)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kratsch, S.: Recent developments in kernelization: a survey. In: Bulletin of the EATCS, no. 113 (2014)Google Scholar
  19. 19.
    Lin, G., Xue, G.: On the terminal Steiner tree problem. Inf. Process. Lett. 84(2), 103–107 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lokshtanov, D., Panolan, F., Ramanujan, M.S., Saurabh, S.: Lossy kernelization. In: Proceedings of 49th STOC, pp. 224–237. ACM (2017)Google Scholar
  21. 21.
    Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6(5), 883–891 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich 4 – Abteilung InformatikUniversität TrierTrierGermany
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  3. 3.Ben Gurion University of the NegevBeershebaIsrael
  4. 4.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

Personalised recommendations