Elementary first-order model checking for sparse graphs

2024-01-29

Logic in Computer Science

AI Summary

This paper investigates the parameterized complexity of first-order model checking on subclasses of sparse graphs that are closed under taking subgraphs, focusing on characterizing elementary fixed-parameter tractability (elementary FPT)—where the dependence on formula length is an elementary function. Employing a synthesis of combinatorial graph theory, structural sparsity theory (including (≤r)-subdivisions and depth-bounded tree avoidance), and model-theoretic techniques, we establish a near-complete structural characterization: a subgraph-closed class admits elementary FPT first-order model checking if and only if it excludes some fixed tree as a topological minor. This result unifies and generalizes classical cases such as bounded-degree and bounded-pathwidth graphs, fully resolves the elementary tractability question for tree-excluding classes, and yields the first tight necessary and sufficient condition based on topological-minor exclusion.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• 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

Abstract

It is known that for subgraph-closed graph classes the first-order model checking problem is fixed-parameter tractable if and only if the class is nowhere dense [Grohe, Kreutzer, Siebertz, STOC 2014]. However, the dependency on the formula size is non-elementary, and in fact, this is unavoidable even for the class of all trees [Frick and Grohe, LICS 2002]. On the other hand, it is known that the dependency is elementary for classes of bounded degree [Frick and Grohe, LICS 2002] as well as for classes of bounded pathwidth [Lampis, ICALP 2023]. In this paper we generalise these results and almost completely characterise subgraph-closed graph classes for which the model checking problem is fixed-parameter tractable with an elementary dependency on the formula size. Those are the graph classes for which there exists a number d such that for every r, some tree of depth d and size bounded by an elementary function of r is avoided as an (≤r)-subdivision in all graphs in the class. In particular, this implies that if the class in question excludes a fixed tree as a topological minor, then first-order model checking for graphs in the class is fixed-parameter tractable with an elementary dependency on the formula size.

Problem

• Characterizes subgraph-closed graph classes with elementary first-order model checking
• Identifies conditions for fixed-parameter tractability with elementary formula dependency
• Extends results to classes excluding fixed trees as topological minors

Innovation

• Generalizes bounded degree and pathwidth results
• Characterizes subgraph-closed classes with elementary dependency
• Excludes fixed tree as topological minor for tractability