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On Approximation Complexity of Metric Dimension Problem

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Combinatorial Algorithms (IWOCA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6460))

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Abstract

We study the approximation complexity of the Metric Dimension problem in bounded degree, dense as well as in general graphs. For the general case, we prove that the Metric Dimension problem is not approximable within \((\!1\!-\!\epsilon\!)\!\ln\!n\) for any \(\epsilon\!>\!0\), unless \(NP\!\subseteq\!DTIME(\!n^{\log\!\log\!n}\!)\), and we give an approximation algorithm which matches the lower bound. Even for bounded degree instances it is APX-hard to determine (compute) the exact value of the metric dimension which we prove by constructing an approximation preserving reduction from the bounded degree Vertex Cover problem.

The special case, in which the underlying graph is superdense turns out to be APX-complete. In particular, we present a greedy constant factor approximation algorithm for these kind of instances and construct a approximation preserving reduction from the bounded degree Dominating Set problem. We also provide first explicit approximation lower bounds for the Metric Dimension problem restricted to dense and bounded degree graphs.

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Authors and Affiliations

  1. Dept. of Computer Science, University of Bonn, Germany

    Mathias Hauptmann, Richard Schmied & Claus Viehmann

Authors
  1. Mathias Hauptmann
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  2. Richard Schmied
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  3. Claus Viehmann
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of London, King’s College, The Strand, WC2R 2LS, London, UK

    Costas S. Iliopoulos

  2. Department of Computing and Software, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, ON, Canada

    William F. Smyth

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Hauptmann, M., Schmied, R., Viehmann, C. (2011). On Approximation Complexity of Metric Dimension Problem. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2010. Lecture Notes in Computer Science, vol 6460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19222-7_15

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  • DOI: https://doi.org/10.1007/978-3-642-19222-7_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-19221-0

  • Online ISBN: 978-3-642-19222-7

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