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Minimizing Corners in Colored Rectilinear Grids

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WALCOM: Algorithms and Computation (WALCOM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14549))

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Abstract

Given a rectilinear grid \(G\), in which cells are either assigned a single color, out of k possible colors, or remain white, can we color white grid cells of \(G\) to minimize the total number of corners of the resulting colored rectilinear polygons in \(G\)? We show how this problem relates to hypergraph visualization, prove that it is NP-hard even for \(k=2\), and present an exact dynamic programming algorithm. Together with a set of simple kernelization rules, this leads to an FPT-algorithm in the number of colored cells of the input. We additionally provide an XP-algorithm in the solution size, and a polynomial \(\mathcal {O}(OPT)\)-approximation algorithm.

This work has been supported by the Vienna Science and Technology Fund (WWTF) under grant [10.47379/ICT19035].

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Acknowledgements

The authors would like to thank anonymous referees for their careful reviews and pointing us to [6].

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Authors and Affiliations

  1. TU Wien, Vienna, Austria

    Thomas Depian, Alexander Dobler & Christoph Kern

  2. TU Eindhoven, Eindhoven, Netherlands

    Jules Wulms

Authors
  1. Thomas Depian
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  2. Alexander Dobler
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  3. Christoph Kern
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  4. Jules Wulms
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Corresponding author

Correspondence to Jules Wulms .

Editor information

Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Ryuhei Uehara

  2. Iwate University, Morioka, Japan

    Katsuhisa Yamanaka

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

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© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

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Depian, T., Dobler, A., Kern, C., Wulms, J. (2024). Minimizing Corners in Colored Rectilinear Grids. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_11

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  • DOI: https://doi.org/10.1007/978-981-97-0566-5_11

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-97-0565-8

  • Online ISBN: 978-981-97-0566-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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