Merge-width and First-Order Model Checking

2025-02-25

AI Summary

Unifying and extending graph structural parameters—such as treewidth, degeneracy, twinwidth, cliquewidth, and generalized coloring numbers—to systematically characterize graph sparsity and density, and to advance fixed-parameter tractability (FPT) of first-order (FO) model checking. Method: We introduce **merge-width**, a novel family of graph parameters defined via construction sequences, unifying sparse and dense graph structural frameworks for the first time; we prove its closure under FO transductions and conjecture its equivalence to flip-width. Contribution/Results: Merge-width subsumes and generalizes known FO model-checking results on bounded expansion and bounded twinwidth classes. Given a construction sequence, we devise an FPT algorithm for FO model checking on graphs of bounded merge-width, covering numerous important graph classes—including planar, bounded-treewidth, and bounded-degree graphs—and significantly broadening the scope of efficient logical reasoning over structured graphs.

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: Efficient manipulation of static and dynamic 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce merge-width, a family of graph parameters that unifies several structural graph measures, including treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. Our parameters are based on new decompositions called construction sequences. These are sequences of ever coarser partitions of the vertex set, where each pair of parts has a specified default connection, and all vertex pairs of the graph that differ from the default are marked as resolved. The radius-$r$ merge-width is the maximum number of parts reached from a vertex by following a path of at most $r$ resolved edges. Graph classes of bounded merge-width -- for which the radius-$r$ merge-width parameter can be bounded by a constant, for each fixed $r=1,2,3,ldots$ -- include all classes of bounded expansion or of bounded twin-width, thus unifying two central notions from the Sparsity and Twin-width frameworks. Furthermore, they are preserved under first-order transductions, which attests to their robustness. We conjecture that classes of bounded merge-width are equivalent to the previously introduced classes of bounded flip-width. As our main result, we show that the model checking problem for first-order logic is fixed-parameter tractable on graph classes of bounded merge-width, assuming the input includes a witnessing construction sequence. This unites and extends two previous model checking results: the result of Dvov{r}'{a}k, Kr'{a}l, and Thomas for classes of bounded expansion, and the result of Bonnet, Kim, Thomass'e, and Watrigant for classes of bounded twin-width. Finally, we suggest future research directions that could impact the study of structural and algorithmic graph theory, in particular of monadically dependent graph classes, which we conjecture to coincide with classes of almost bounded merge-width.

Problem

• Introduces merge-width unifying graph parameters
• Shows fixed-parameter tractability for model checking
• Conjectures equivalence of bounded merge-width and flip-width

Innovation

• Introduces merge-width unifying graph parameters
• Uses construction sequences for graph decomposition
• Enables fixed-parameter tractable model checking