🤖 AI Summary
This paper investigates the tractability boundary and computational complexity of first-order (FO) model checking on tournaments. Method: Integrating twin-width theory, NIP model theory, and structural graph theory, the authors develop novel combinatorial and logical characterizations. Contribution/Results: They establish the first FPT/AW[*]-hard dichotomy for FO model checking on tournaments; prove the equivalence of four properties—NIPness, bounded twin-width, exponential growth restriction, and FO tractability; provide three obstruction-based characterizations of twin-width; and design the first polynomial-time twin-width approximation algorithm leveraging linear orders. Furthermore, they extend these results to dense directed graphs with bounded stability number, thereby unifying the precise correspondence between FO tractability and combinatorial/model-theoretic structural properties.
📝 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 $ extrm{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 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 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.
