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Upper Domination: Complexity and Approximation

  • Cristina Bazgan
  • Ljiljana Brankovic
  • Katrin Casel
  • Henning Fernau
  • Klaus Jansen
  • Kim-Manuel Klein
  • Michael Lampis
  • Mathieu Liedloff
  • Jérôme Monnot
  • Vangelis Th. Paschos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

We consider Upper Domination, the problem of finding a maximum cardinality minimal dominating set in a graph. We show that this problem does not admit an \(n^{1-\epsilon }\) approximation for any \(\epsilon >0\), making it significantly harder than Dominating Set, while it remains hard even on severely restricted special cases, such as cubic graphs (APX-hard), and planar subcubic graphs (NP-hard). We complement our negative results by showing that the problem admits an \(O(\varDelta )\) approximation on graphs of maximum degree \(\varDelta \), as well as an EPTAS on planar graphs. Along the way, we also derive essentially tight \(n^{1-\frac{1}{d}}\) upper and lower bounds on the approximability of the related problem Maximum Minimal Hitting Set on d-uniform hypergraphs, generalising known results for Maximum Minimal Vertex Cover.

Notes

Acknowledgements

We thank anonymous referees for helpful comments. We also thank our colleagues David Manlove and Daniel Meister for discussions on upper domination. Part of this research was supported by the DFG, grant FE 560/6-1.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Cristina Bazgan
    • 1
  • Ljiljana Brankovic
    • 2
    • 3
  • Katrin Casel
    • 3
  • Henning Fernau
    • 3
  • Klaus Jansen
    • 4
  • Kim-Manuel Klein
    • 4
  • Michael Lampis
    • 1
  • Mathieu Liedloff
    • 5
  • Jérôme Monnot
    • 1
  • Vangelis Th. Paschos
    • 1
  1. 1.Institut Universitaire de FranceUniversité Paris-Dauphine, PSL Research University, CNRS, UMR 7243ParisFrance
  2. 2.The University of NewcastleCallaghanAustralia
  3. 3.Universität TrierTrierGermany
  4. 4.Universität Kiel, Institut für InformatikKielGermany
  5. 5.Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022OrléansFrance

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