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Approximating Modular Decomposition Is Hard

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Algorithms and Discrete Applied Mathematics (CALDAM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12016))

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Abstract

In order to understand underlying structural regularities in a graph, a basic and useful technique, known as modular decomposition, looks for subsets of vertices that have the exact same neighbourhood to the outside. These are known as modules and there exist linear-time algorithms to find them. This notion however is too strict, especially when dealing with graphs that arise from real world data. This is why it is important to relax this condition by allowing some noise in the data. However, generalizing modular decomposition is far from being obvious since most of the proposals lose the algebraic properties of modules and therefore most of the nice algorithmic consequences. In this paper we introduce the notion of \(\epsilon \)-module which seems to be a good compromise that maintains some of the algebraic structure. Among the main results in the paper, we show that minimal \(\epsilon \)-modules can be computed in polynomial time, on the other hand for maximal \(\epsilon \)-modules it is already NP-hard to compute if a graph admits an 1-parallel decomposition, i.e. one step of decomposition of \(\epsilon \)-module with \(\epsilon =1\).

This work is supported by the ANR-France Project Hosigra (ANR-17-CE40-0022).

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Notes

  1. 1.

    We use \(\epsilon \) to denote small error, despite being greater than 1.

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Acknowledgments

The authors wish to thank anonymous reviewers for their helpful comments.

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Authors and Affiliations

  1. IRIF, UMR 8243 CNRS & Paris University, Paris, France

    Michel Habib & Mengchuan Zou

  2. Department of Computer Science, University of Toronto, Toronto, ON, Canada

    Lalla Mouatadid

  3. Gang Project, Inria Paris, Paris, France

    Michel Habib & Mengchuan Zou

Authors
  1. Michel Habib
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  2. Lalla Mouatadid
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  3. Mengchuan Zou
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Corresponding author

Correspondence to Michel Habib .

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Editors and Affiliations

  1. University of Kerala, Thiruvananthapuram, India

    Manoj Changat

  2. Indian Statistical Institute, Kolkata, India

    Sandip Das

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Habib, M., Mouatadid, L., Zou, M. (2020). Approximating Modular Decomposition Is Hard. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_5

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  • DOI: https://doi.org/10.1007/978-3-030-39219-2_5

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-39219-2

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