On problems without polynomial kernels

  • Authors:

  • Hans L. Bodlaender;Rodney G. Downey;Michael R. Fellows;Danny Hermelin

  • Affiliations:

  • Department of Information and Computing Sciences, Utrecht University, 3508 TB Utrecht 80.089, Netherlands;School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, Wellington 600, New Zealand;The University of Newcastle, Callaghan NSW 2308, Australia;Department of Computer Science, The University of Haifa, Mount Carmel, Haifa 31905, Israel

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2009

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Abstract

Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.