Twin-width I: tractable FO model checking

Edouard Bonnet (1) , Eun Jung Kim (2) , Stéphan Thomassé (1) , Rémi Watrigant (1)

1 MC2 - Modèles de calcul, Complexité, Combinatoire
2 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_t$-free unit $d$-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions, witness that the twin-width is at most $d$. We show that FO model checking, that is deciding if a given first-order formula $\phi$ evaluates to true for a given binary structure $G$ on a domain $D$, is FPT in $|\phi|$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$-contraction sequence for $G$, our algorithm runs in time $f(d,|\phi|) \cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarsk\'y et al. [FOCS '15].

Domaines: Informatique [cs]

Edouard Bonnet, Eun Jung Kim, Stéphan Thomassé, Rémi Watrigant. Twin-width I: tractable FO model checking. FOCS 2020, Nov 2020, online, United States. ⟨hal-03107581⟩