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Complexity of Maximum Fixed Point Problem in Boolean Networks

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Computing with Foresight and Industry (CiE 2019)

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Abstract

A Boolean network (BN) with n components is a discrete dynamical system described by the successive iterations of a function \(f:\{{ \texttt {0}},{ \texttt {1}}\}^n \rightarrow \{{ \texttt {0}},{ \texttt {1}}\}^n\). This model finds applications in biology, where fixed points play a central role. For example in genetic regulation they correspond to cell phenotypes. In this context, experiments reveal the existence of positive or negative influences among components: component i has a positive (resp. negative) influence on component j, meaning that j tends to mimic (resp. negate) i. The digraph of influences is called signed interaction digraph (SID), and one SID may correspond to multiple BNs. The present work opens a new perspective on the well-established study of fixed points in BNs. Biologists discover the SID of a BN they do not know, and may ask: given that SID, can it correspond to a BN having at least k fixed points? Depending on the input, this problem is in \( \textsf {P}\) or complete for \(\textsf {NP}\), \(\textsf {NP}^\textsf {\#P}\) or \(\textsf {NEXPTIME}\).

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Acknowledgments

The authors would like to thank for their support the Young Researcher project ANR-18-CE40-0002-01 “FANs”, project ECOS-CONICYT C16E01, and project STIC AmSud CoDANet 19-STIC-03 (Campus France 43478PD).

Author information

Authors and Affiliations

  1. Aix-Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France

    Florian Bridoux, Nicolas Durbec & Kevin Perrot

  2. Laboratoire I3S, CNRS, Université Côte d’Azur, Nice, France

    Adrien Richard

  3. CMM, UMI CNRS 2807, Universidad de Chile, Santiago, Chile

    Adrien Richard

Authors
  1. Florian Bridoux
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  2. Nicolas Durbec
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  3. Kevin Perrot
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  4. Adrien Richard
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Corresponding author

Correspondence to Florian Bridoux .

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Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Bridoux, F., Durbec, N., Perrot, K., Richard, A. (2019). Complexity of Maximum Fixed Point Problem in Boolean Networks. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_12

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22995-5

  • Online ISBN: 978-3-030-22996-2

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