π€ AI Summary
This study investigates the fixed-parameter tractability (FPT) of first-order model checking when parameterized by the number of variables in the logical formula. Focusing on monotone and hereditary graph classes, the paper provides the first systematic characterization of graph classes that admit FPT algorithms: a complete characterization is established for monotone classes, while an approximate characterization is given for hereditary classes. By integrating techniques from parameterized complexity theory, first-order model checking, and structural graph analysis, the work precisely delineates the boundary of FPT tractability under variable-number parameterization, thereby significantly advancing the understanding of the parameterized complexity landscape of first-order model checking.
π Abstract
The first-order (FO) model checking problem asks, given an FO sentence $Ο$ and a graph $G$, whether $G$ is a model of $Ο$. This problem is known to be $\mathsf{AW[*]}$-hard when parameterized by the quantifier rank of the formula. A classical algorithm decides this problem in XP-time parameterized by the number of variables in the formula.
Due to $\mathsf{AW[*]}$-hardness, it is natural to ask about the complexity of the problem when restricted to some well-behaved class of graphs. There are many results describing graph classes $\mathcal{C}$ such that the FO model checking problem restricted to $\mathcal{C}$ admits an $\mathsf{FPT}$-time algorithm when parameterized by the quantifier rank of the formula.
Parameterization by the quantifier rank is significantly more restrictive than parameterization by the number of variables. We investigate the graph classes $\mathcal{C}$ for which the FO model checking problem restricted to $\mathcal{C}$ admits an $\mathsf{FPT}$-time algorithm when parameterized by the number of variables in the formula. We characterize these classes in the monotone setting, and prove a slightly weaker result in the hereditary setting.