Complexity of Maximum Fixed Point Problem in Boolean Networks

  • Florian BridouxEmail author
  • Nicolas Durbec
  • Kevin Perrot
  • Adrien Richard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11558)


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 \(\textsf {NEXPTIME}\).



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).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Florian Bridoux
    • 1
    Email author
  • Nicolas Durbec
    • 1
  • Kevin Perrot
    • 1
  • Adrien Richard
    • 2
    • 3
  1. 1.Aix-Marseille Université, Université de Toulon, CNRS, LISMarseilleFrance
  2. 2.Laboratoire I3S, CNRSUniversité Côte d’AzurNiceFrance
  3. 3.CMM, UMI CNRS 2807, Universidad de ChileSantiagoChile

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