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Parameterized Complexity of Safe Set

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Algorithms and Complexity (CIAC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11485))

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Abstract

In this paper we study the problem of finding a small safe set S in a graph G, i.e. a non-empty set of vertices such that no connected component of G[S] is adjacent to a larger component in \(G - S\). We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth \(\mathsf {pw}\) and cannot be solved in time \(n^{o(\mathsf {pw})}\) unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number \(\mathsf {vc}\) unless \(\mathrm {PH} = \varSigma ^{\mathrm {p}}_{3}\), but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity \(\mathsf {nd}\), and (4) it can be solved in time \(n^{f(\mathsf {cw})}\) for some double exponential function f where \(\mathsf {cw}\) is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.

Partially supported by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA) Project GRAPA 38593YJ, and by JSPS/MEXT KAKENHI Grant Numbers JP24106004, JP17H01698, JP18K11157, JP18K11168, JP18K11169, JP18H04091, 18H06469.

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Notes

  1. 1.

    The \(O^*(\cdot )\) notation omits the polynomial dependency on the input size.

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Author information

Authors and Affiliations

  1. The University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan

    Rémy Belmonte

  2. Chuo University, Bunkyo-ku, Tokyo, 112-8551, Japan

    Tesshu Hanaka

  3. Université Paris-Dauphine, PSL University, CNRS, LAMSADE, Paris, France

    Ioannis Katsikarelis & Michael Lampis

  4. Nagoya University, Nagoya, 464-8601, Japan

    Hirotaka Ono

  5. Kumamoto University, Kumamoto, 860-8555, Japan

    Yota Otachi

Authors
  1. Rémy Belmonte
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  2. Tesshu Hanaka
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  3. Ioannis Katsikarelis
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  4. Michael Lampis
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  5. Hirotaka Ono
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  6. Yota Otachi
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Corresponding author

Correspondence to Ioannis Katsikarelis .

Editor information

Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Belmonte, R., Hanaka, T., Katsikarelis, I., Lampis, M., Ono, H., Otachi, Y. (2019). Parameterized Complexity of Safe Set. In: Heggernes, P. (eds) Algorithms and Complexity. CIAC 2019. Lecture Notes in Computer Science(), vol 11485. Springer, Cham. https://doi.org/10.1007/978-3-030-17402-6_4

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  • DOI: https://doi.org/10.1007/978-3-030-17402-6_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-17401-9

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