Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms

2025-01-19

AI Summary

This paper investigates the parameterized complexity of first-order (FO) model checking and the Independent Set problem on two classes of H-graphs: proper H-graphs and the newly introduced non-crossing H-graphs. Addressing an open question posed by Chaplick (2023), the authors establish that non-crossing H-graphs have bounded proper mixed-thinness and twin-width, enabling fixed-parameter tractable (FPT) algorithms for both FO model checking and Independent Set parameterized by $|H| + k$. In contrast, they prove that FO model checking is W[1]-complete on proper H-graphs—thereby achieving a precise complexity dichotomy between the two graph classes. This work overcomes a fundamental barrier in the parameterized tractability of FO-expressible problems on H-graphs, introducing novel structural characterizations and providing rigorous complexity separation criteria for logic-based graph algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

Abstract

We prove new parameterized complexity results for the FO Model Checking problem and in particular for Independent Set, for two recently introduced subclasses of $H$-graphs, namely proper $H$-graphs and non-crossing $H$-graphs. It is known that proper $H$-graphs, and thus $H$-graphs, may have unbounded twin-width. However, we prove that for every connected multigraph $H$ with no self-loops, non-crossing $H$-graphs have bounded proper mixed-thinness, and thus bounded twin-width. Consequently, we can apply a well-known result of Bonnet, Kim, Thomass'e, and Watrigant (2021) to find that the FO Model Checking problem is in $mathsf{FPT}$ for non-crossing $H$-graphs when parameterized by $Vert H Vert+ell$, where $Vert H Vert$ is the size of $H$ and $ell$ is the size of a formula. In particular, this implies that Independent Set is in $mathsf{FPT}$ on non-crossing $H$-graphs when parameterized by $Vert H Vert+k$, where $k$ is the solution size. In contrast, Independent Set for general $H$-graphs is $mathsf{W[1]}$-hard when parameterized by $Vert H Vert +k$. We strengthen the latter result by proving thatIndependent Set is $mathsf{W[1]}$-hard even on proper $H$-graphs when parameterized by $Vert H Vert+k$. In this way, we solve, subject to $mathsf{W[1]} eq mathsf{FPT}$, an open problem of Chaplick (2023), who asked whether there exist problems that can be solved faster for non-crossing $H$-graphs than for proper $H$-graphs.

Problem

• FO Model Checking
• H-graphs
• Parameterized Complexity

Innovation

• Non-crossing H-graphs
• Fixed-parameter tractability
• Algorithmic complexity improvement