A parameterized route to exact puzzles: breaking the 2n-barrier for irredundance (Extended Abstract)

  • Authors:

  • Daniel Binkele-Raible;Ljiljana Brankovic;Henning Fernau;Joachim Kneis;Dieter Kratsch;Alexander Langer;Mathieu Liedloff;Peter Rossmanith

  • Affiliations:

  • FB 4—Abteilung Informatik, Universität Trier, Trier, Germany;The University of Newcastle, Callaghan, Australia;FB 4—Abteilung Informatik, Universität Trier, Trier, Germany;Department of Computer Science, RWTH Aachen University, Germany;Laboratoire d'Informatique Théorique et Appliquée, Université Paul Verlaine - Metz, Metz Cedex 01, France;Department of Computer Science, RWTH Aachen University, Germany;Laboratoire d'Informatique Fondamentale d'Orléans, Université d'Orléans, Orléans Cedex 2, France;Department of Computer Science, RWTH Aachen University, Germany

  • Venue:
  • CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
  • Year:
  • 2010

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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.