Lower Bounds for Kernelizations and Other Preprocessing Procedures

  • Authors:

  • Yijia Chen;Jörg Flum;Moritz Müller

  • Affiliations:

  • BASICS, Department of Computer Science, Shanghai Jiaotong University, Shanghai, China 200030;Abteilung für Mathematische Logik, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany 79104;Abteilung für Mathematische Logik, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany 79104

  • Venue:
  • CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
  • Year:
  • 2009

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Abstract

We first present a method to rule out the existence of strong polynomial kernelizations of parameterized problems under the hypothesis P $\ne$ NP. This method is applicable, for example, to the problem Sat parameterized by the number of variables of the input formula. Then we obtain improvements of related results in [1,6] by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming that PH $\ne \Sigma^{\rm {P}}_3$, i.e., that the polynomial hierarchy does not collapse to its third level, we show that every parameterized problem with a "linear OR" and with NP-hard underlying classical problem does not have polynomial reductions to itself that assign to every instance x with parameter k an instance y with |y | = k O (1)·|x |1 *** *** (here *** is any given real number greater than zero).