A Parameterized Route to Exact Puzzles: Breaking the 2n-Barrier for Irredundance

(Extended Abstract)
  • Daniel Binkele-Raible
  • Ljiljana Brankovic
  • Henning Fernau
  • Joachim Kneis
  • Dieter Kratsch
  • Alexander Langer
  • Mathieu Liedloff
  • Peter Rossmanith
Conference paper

DOI: 10.1007/978-3-642-13073-1_28

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)
Cite this paper as:
Binkele-Raible D. et al. (2010) A Parameterized Route to Exact Puzzles: Breaking the 2n-Barrier for Irredundance. In: Calamoneri T., Diaz J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg

Abstract

The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2n) enumeration barrier. We solve this open problem by devising parameterized algorithms for the duals of the natural parameterizations of the problems with running times faster than \(\mathcal{O}^*(4^{k})\). For example, we present an algorithm running in time \(\mathcal{O}^*(3.069^{k}))\) for determining whether IR(G) is at least n − k. Although the corresponding problem has been shown to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Furthermore, these seem to be the first examples of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as exact exponential-time algorithms fails.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Daniel Binkele-Raible
    • 1
  • Ljiljana Brankovic
    • 2
  • Henning Fernau
    • 1
  • Joachim Kneis
    • 3
  • Dieter Kratsch
    • 4
  • Alexander Langer
    • 3
  • Mathieu Liedloff
    • 5
  • Peter Rossmanith
    • 3
  1. 1.FB 4—Abteilung InformatikUniversität TrierTrierGermany
  2. 2.The University of NewcastleCallaghanAustralia
  3. 3.Department of Computer ScienceRWTH Aachen UniversityGermany
  4. 4.Laboratoire d’Informatique Théorique et AppliquéeUniversité Paul Verlaine - MetzMetz Cedex 01France
  5. 5.Laboratoire d’Informatique Fondamentale d’OrléansUniversité d’OrléansOrléans Cedex 2France

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