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Bounding the Number of Minimal Dominating Sets: A Measure and Conquer Approach

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Book cover Algorithms and Computation (ISAAC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3827))

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Abstract

We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697n, thus improving on the trivial \(\mathcal{O}(2^{n}/\sqrt{n})\) bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an \(\mathcal{O}(1.7697^{n})\) listing algorithm.

Based on this result, we derive an \(\mathcal{O}(2.8805^{n})\) algorithm for the domatic number problem, and an \(\mathcal{O}(1.5780^{n})\) algorithm for the minimum-weight dominating set problem. Both algorithms improve over the previous algorithms.

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    Fedor V. Fomin, Artem V. Pyatkin & Alexey A. Stepanov

  2. Dipartimento di Informatica, Università di Roma “La Sapienza”, Via Salaria 113, 00198, Roma, Italy

    Fabrizio Grandoni

Authors
  1. Fedor V. Fomin
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  2. Fabrizio Grandoni
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  3. Artem V. Pyatkin
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  4. Alexey A. Stepanov
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Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon , Hong Kong

    Xiaotie Deng

  2. Department of Computer Science, University of Texas at Dallas, EE/CS Building, 75083, Richardson, TX, USA

    Ding-Zhu Du

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© 2005 Springer-Verlag Berlin Heidelberg

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Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A. (2005). Bounding the Number of Minimal Dominating Sets: A Measure and Conquer Approach. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_58

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  • DOI: https://doi.org/10.1007/11602613_58

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30935-2

  • Online ISBN: 978-3-540-32426-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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