Structural Indexing of Relational Databases for the Evaluation of Free-Connex Acyclic Conjunctive Queries

We present an index structure to boost the evaluation of free-connex acyclic conjunctive queries (fc-ACQs) over relational databases. The main ingredient of the index associated with a given database $D$ is an auxiliary database $D_{col}$. Our main result states that for any fc-ACQ $Q$ over $D$, we can count the number of answers of $Q$ or enumerate them with constant delay after a preprocessing phase that takes time linear in the size of $D_{col}$. Unlike previous indexing methods based on values or order (e.g., B+ trees), our index is based on structural symmetries among tuples in a database, and the size of $D_{col}$ is related to the number of colors assigned to $D$ by Scheidt and Schweikardt's "relational color refinement" (2025). In the particular case of graphs, this coincides with the minimal size of an equitable partition of the graph. For example, the size of $D_{col}$ is logarithmic in the case of binary trees and constant for regular graphs. Even in the worst-case that $D$ has no structural symmetries among tuples at all, the size of $D_{col}$ is still linear in the size of $D$. Given that the size of $D_{col}$ is bounded by the size of $D$ and can be much smaller (even constant for some families of databases), our index is the first foundational result on indexing internal structural symmetries of a database to evaluate all fc-ACQs with performance potentially strictly smaller than the database size.