International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 287-298 | Cite as

Polynomial Kernels for Weighted Problems

  • Michael Etscheid
  • Stefan Kratsch
  • Matthias Mnich
  • Heiko Röglin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)

Abstract

Kernelization is a formalization of efficient preprocessing for \(\mathsf {NP}\)-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number n of items.

We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and Végh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Michael Etscheid
    • 1
  • Stefan Kratsch
    • 1
  • Matthias Mnich
    • 1
  • Heiko Röglin
    • 1
  1. 1.Institut für InformatikUniversität BonnBonnGermany

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