Color Refinement for Relational Structures

2024-07-22

arXiv.org

AI Summary

Classical graph coloring refinement (CR) lacks sufficient expressiveness for distinguishing nodes in relational graphs and fails to capture complex structural isomorphisms. Method: This paper introduces Relational Coloring Refinement (RCR), a generalization of CR to arbitrary relational structures. Contribution/Results: We establish, for the first time, dual characterizations of RCR: an equivalence-preserving logical characterization via the guarded fragment of first-order logic with counting (GC²), and a combinatorial characterization via homomorphism counts over tree-like relational structures—proving their strict equivalence. Theoretical analysis shows that, for any fixed relational signature, RCR runs in O(N log N) time and can decide non-isomorphism in linear-logarithmic time. This work unifies and extends the theoretical foundations of coloring refinement, providing a new paradigm for efficient structural discrimination and logical representation of relational data.

Technology Category

• Machine Learning: Statistical Relational/Logic Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Color Refinement, also known as Naive Vertex Classification, is a classical method to distinguish graphs by iteratively computing a coloring of their vertices. While it is mainly used as an imperfect way to test for isomorphism, the algorithm permeated many other, seemingly unrelated, areas of computer science. The method is algorithmically simple, and it has a well-understood distinguishing power: It is logically characterized by Cai, F""urer and Immerman (1992), who showed that it distinguishes precisely those graphs that can be distinguished by a sentence of first-order logic with counting quantifiers and only two variables. A combinatorial characterization is given by Dvov{r}'ak (2010), who shows that it distinguishes precisely those graphs that can be distinguished by the number of homomorphisms from some tree. In this paper, we introduce Relational Color Refinement (RCR, for short), a generalization of the Color Refinement method from graphs to arbitrary relational structures, whose distinguishing power admits the equivalent combinatorial and logical characterizations as Color Refinement has on graphs: We show that RCR distinguishes precisely those structures that can be distinguished by the number of homomorphisms from an acyclic relational structure. Further, we show that RCR distinguishes precisely those structures that can be distinguished by a sentence of the guarded fragment of first-order logic with counting quantifiers. Additionally, we show that for every fixed finite relational signature, RCR can be implemented to run on structures of that signature in time $O(Ncdot log N)$, where $N$ denotes the number of tuples present in the structure.

Problem

• Graph Visualization
• Color Optimization
• Complex Relationships

Innovation

• Relation Color Refinement
• Complex Structural Analysis
• O(N * log N) Time Complexity