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First Order Logic and Twin-Width in Tournaments

Authors Colin Geniet , Stéphan Thomassé

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Author Details

Colin Geniet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

Funding

The authors were partially supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014-01) and DIGRAPHS (ANR-19-CE48-0013-01).

Acknowledgements

The authors would like to thank Édouard Bonnet for stimulating discussions on this topics, and Szymon Toruńczyk for helpful explanations on the notion of restrained classes.

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Colin Geniet and Stéphan Thomassé. First Order Logic and Twin-Width in Tournaments. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.53

Abstract

We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments T, first-order model checking either is fixed parameter tractable, or is AW[*]-hard. This dichotomy coincides with the fact that T has either bounded or unbounded twin-width, and that the growth of T is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: T has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs.
The key for these results is a polynomial time algorithm which takes as input a tournament T and computes a linear order < on V(T) such that the twin-width of the birelation (T, <) is at most some function of the twin-width of T. Since approximating twin-width can be done in FPT time for an ordered structure (T, <), this provides a FPT approximation of twin-width for tournaments.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph enumeration
Keywords
  • Tournaments
  • twin-width
  • first-order logic
  • model checking
  • NIP
  • small classes

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