Efficient reversal of transductions of sparse graph classes

2026-01-21

AI Summary

This work investigates the efficient approximate inversion of first-order definable graph transductions where the source graphs belong to sparse classes. For graph classes with structurally bounded expansion, we design an O(n⁴)-time algorithm that, given an input graph G, reconstructs a vertex-colored graph H of bounded expansion such that G can be recovered from H via a first-order interpretation. Our result resolves an open problem posed by Gajarský et al., significantly weakening the required conditions for invertible transductions to merely unary stability and inherent linear neighborhood complexity. As a consequence, we obtain a novel equivalent characterization of structurally bounded expansion classes, thereby establishing a deep connection between structural graph properties and logical definability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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: Bridging structured and unstructured data

Abstract

(First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarsk\'y et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion.

Problem

• transductions
• sparse graph classes
• bounded expansion
• monadic stability
• neighborhood complexity

Innovation

• first-order transductions
• structurally bounded expansion
• monadic stability
• neighborhood complexity
• graph sparsity