Breaking the 2n-barrier for Irredundance: Two lines of attack

  • Authors:

  • Daniel Binkele-Raible;Ljiljana Brankovic;Marek Cygan;Henning Fernau;Joachim Kneis;Dieter Kratsch;Alexander Langer;Mathieu Liedloff;Marcin Pilipczuk;Peter Rossmanith;Jakub Onufry Wojtaszczyk

  • Affiliations:

  • FB 4 - Abteilung Informatik, Universität Trier, Germany;School of Electrical Engineering and Computer Science, The University of Newcastle, Australia;Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland;FB 4 - Abteilung Informatik, Universität Trier, Germany;Dept. of Computer Science, RWTH Aachen University, Germany;Laboratoire dInformatique Théorique et Appliquée, Université Paul Verlaine Metz, France;Dept. of Computer Science, RWTH Aachen University, Germany;Laboratoire dInformatique Fondamentale dOrléans, Université dOrléans, France;Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland;Dept. of Computer Science, RWTH Aachen University, Germany;Institute of Mathematics, Polish Academy of Sciences, Poland

  • Venue:
  • Journal of Discrete Algorithms
  • Year:
  • 2011

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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 the domination and independence numbers and have numerous relations to other graph parameters. It has been an open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time faster than the trivial @Q(2^n@?poly(n)) enumeration, also called the 2^n-barrier. The main contributions of this article are exact exponential-time algorithms breaking the 2^n-barrier for irredundance. We establish algorithms with running times of O^@?(1.99914^n) for computing ir(G) and O^@?(1.9369^n) for computing IR(G). Both algorithms use polynomial space. The first algorithm uses a parameterized approach to obtain (faster) exact algorithms. The second one is based, in addition, on a reduction to the Maximum Induced Matching problem providing a branch-and-reduce algorithm to solve it.