Color Refinement for Relational Structures

TL;DR

This paper generalizes the classical Color Refinement algorithm from graphs to arbitrary relational structures, providing logical and combinatorial characterizations of its distinguishing power and an efficient implementation.

Contribution

It introduces Relational Color Refinement (RCR), extending the method to relational structures with equivalent logical and combinatorial characterizations and a near-linear runtime.

Findings

RCR distinguishes structures by homomorphisms from acyclic structures.

RCR characterizes structures definable by guarded fragment logic with counting.

RCR runs in O(N log N) time for fixed signatures.

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 Dvo\v{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 $O(N\cdot \log N)$, where $N$ denotes the number of tuples present in the structure.