Betweenness parameterized above tight lower bound

  • Authors:

  • Gregory Gutin;Eun Jung Kim;Matthias Mnich;Anders Yeo

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK;Technische Universiteit Eindhoven, Eindhoven, The Netherlands;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2010

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Abstract

We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the Betweenness problem parameterized above its tight lower bound, which is stated as follows. For a set V of variables and set C of constraints ''v"i is between v"j and v"k'', decide whether there is a bijection from V to the set {1,...,|V|} satisfying at least |C|/3+@k of the constraints in C. Our result solves an open problem attributed to Benny Chor in Niedermeier's monograph ''Invitation to Fixed-Parameter Algorithms''. The betweenness problem is of interest in molecular biology. An approach developed in this paper can be used to determine parameterized complexity of a number of other optimization problems on permutations parameterized above or below tight bounds.