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International Conference on Current Trends in Theory and Practice of Informatics

SOFSEM 2021: SOFSEM 2021: Theory and Practice of Computer Science pp 89–103Cite as

Balanced Independent and Dominating Sets on Colored Interval Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12607))

Abstract

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely f-Balanced Independent Set (f-BIS) and f-Balanced Dominating Set (f-BDS). Let \(G=(V,E)\) be an interval graph with a color assignment function \({{\,\mathrm{\gamma }\,}}:V \rightarrow \{1,\ldots ,k\}\) that maps all vertices in G onto k colors. A subset of vertices \(S\subseteq V\) is called f-balanced if S contains f vertices from each color class. In the f-BIS and f-BDS problems, the objective is to compute an independent set or a dominating set that is f-balanced. We show that both problems are NP-complete even on proper interval graphs. For the BIS problem on interval graphs, we design two FPT algorithms, one parameterized by (f, k) and the other by the vertex cover number of G. Moreover, for an optimization variant of BIS on interval graphs, we present a polynomial time approximation scheme (PTAS) and an \(O(n\log n)\) time 2-approximation algorithm.

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Notes

  1. 1.

    We use the term color assignment instead of vertex coloring to avoid any confusion with the general notion of vertex coloring; in particular, a color assignment \({{\,\mathrm{\gamma }\,}}\) can map adjacent vertices to the same color.

  2. 2.

    FPT is the class of parameterized problems that can be solved in time \(O(g(k)n^{O(1)})\) for input size n, parameter k, and some computable function g.

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Acknowledgements

This work was supported by the Austrian Science Fund (FWF) under grant P31119.

Author information

Authors and Affiliations

  1. Université libre de Bruxelles, Brussels, Belgium

    Sujoy Bhore

  2. Geoinformation Group, University of Bonn, Bonn, Germany

    Jan-Henrik Haunert

  3. Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Fabian Klute

  4. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Guangping Li & Martin Nöllenburg

Authors
  1. Sujoy Bhore
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  2. Jan-Henrik Haunert
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  3. Fabian Klute
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  4. Guangping Li
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  5. Martin Nöllenburg
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Corresponding author

Correspondence to Guangping Li .

Editor information

Editors and Affiliations

  1. Charles University, Prague, Czech Republic

    Tomáš Bureš

  2. University of Bergamo, Bergamo, Italy

    Riccardo Dondi

  3. Free University of Bozen-Bolzano, Bolzano, Italy

    Johann Gamper

  4. University of Genoa, Genoa, Italy

    Giovanna Guerrini

  5. University of Wroclaw, Wroclaw, Poland

    Tomasz Jurdziński

  6. Free University of Bozen-Bolzano, Bolzano, Italy

    Claus Pahl

  7. University Paris Dauphine, Paris, France

    Florian Sikora

  8. University of Liverpool, Liverpool, UK

    Prudence W.H. Wong

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Bhore, S., Haunert, JH., Klute, F., Li, G., Nöllenburg, M. (2021). Balanced Independent and Dominating Sets on Colored Interval Graphs. In: Bureš, T., et al. SOFSEM 2021: Theory and Practice of Computer Science. SOFSEM 2021. Lecture Notes in Computer Science(), vol 12607. Springer, Cham. https://doi.org/10.1007/978-3-030-67731-2_7

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  • DOI: https://doi.org/10.1007/978-3-030-67731-2_7

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