Twin-width I: tractable FO model checking (2004.14789v3)

Abstract: 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].

• The paper introduces twin-width and shows that FO model checking becomes tractable in linear FPT time for graphs with bounded twin-width.
• It employs contraction sequences that maintain a low red degree, broadening traditional measures like treewidth to dense graph settings.
• The study applies twin-width to classes such as bounded rank-width graphs, grids, and pattern-avoiding permutations, advancing both theory and algorithm design.

An Examination of Twin-width and Its Applications in Tractable First-order Model Checking

The paper "Twin-width I: tractable FO model checking" introduces a novel graph parameter named twin-width, inspired by width invariant concepts like treewidth and VC-dimension, but adaptable to broader applications in dense graph settings where other parameters may not perform well. This paper presents twin-width as a foundational measure that influences computational tractability, particularly for FO model checking.

Definitions and Concepts

Twin-width: Twin-width is defined based on graph sequences known as contraction sequences, where graphs are simplified iteratively while ensuring the red degree, representing errors or uncertainties introduced during transformations, remains below a threshold $d$. The twin-width of a graph is the minimal $d$ for which such a sequence exists.

Contraction Sequence: It is an iterative process of simplifying a graph through vertex contractions without significantly increasing complexity, evidential through low red degree vertices.

Central Results and Implications

• Bounded Twin-width Classes: The paper establishes that certain classes, including bounded rank-width graphs, grids, permutations avoiding fixed patterns, bounded width posets, and $K_t$-minor free graphs, have bounded twin-width. This implies these classes allow tractable solutions for FO model checking problems.
• Operational Extension: Twin-width is preserved under graph complementation and taking induced subgraphs, and it remains bounded when adding apex vertices or applying first-order interpretations and transductions.
• Algorithmic Applications: The key algorithmic result is that given a graph and a sequence proving its twin-width is bounded, FO model checking can be performed in linear FPT time. This includes applications to problems like $k$-Independent Set, $k$-Dominating Set, and diameter computations for bounded twin-width graphs.

Technical Contributions

Grid Minor Theorem for Twin-width: This theorem parallels the classical relationship between treewidth and grid minors, offering a characterization in terms of mixed minors in matrices. Graphs with bounded twin-width can be recast through a specific vertex ordering that prevents the existence of large mixed minors in their adjacency matrices, extensively broadening potential application fields.

Implications for Theoretical Computer Science: Twin-width offers a framework that unifies known tractability results for FO model checking across diverse graph classes without relying on existing dense graph techniques. The work provides pathways for solving previously intractable problems and suggests a direction for understanding the complexity of classes beyond polynomial expansion or bounded expansion frameworks.

Future Developments in AI and Computation

Given twin-width's wide applicability, its further paper could influence developments in algorithm design for AI, particularly in learning and reasoning tasks involving large datasets modeled as graphs. The paper suggests a potential for extending twin-width concepts to matrix and hypergraph settings, broadening the scope of combinatorial parameterizations for AI applications.

Overall, the concept of twin-width, as introduced in this paper, sets a foundational stone for future investigations into graph complexity and algorithm design, finding a delicate balance between theoretical elegance and practical algorithmic utility in areas such as AI and machine learning models, for which efficient computation and verification of properties on graph-structured data are essential.