First Order Logic and Twin-Width in Tournaments and Dense Oriented Graphs

TL;DR

This paper characterizes when first-order model checking on tournaments is tractable, linking it to twin-width and growth rates, and provides algorithms for computing and approximating twin-width in these structures.

Contribution

It establishes a dichotomy for hereditary classes of tournaments based on twin-width and growth, and introduces polynomial-time algorithms for computing and approximating twin-width.

Findings

First-order model checking is either fixed parameter tractable or hard, depending on twin-width.

Bounded twin-width classes coincide with NIP classes of tournaments.

A polynomial-time algorithm computes a linear order with bounded twin-width approximation.

Abstract

We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments $\mathcal T$, first-order model checking is either fixed parameter tractable or $\textrm{AW}[*]$-hard. This dichotomy coincides with the fact that $\mathcal T$ has either bounded or unbounded twin-width, and that the growth of $\mathcal T$ 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: $\mathcal T$ 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 that 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 polynomial time for an ordered structure $(T,<)$, this provides a polynomial time approximation of twin-width for tournaments. Our results extend to oriented graphs with stable sets of bounded size, which may also be augmented by arbitrary binary relations.