The Complexity of Dependency Detection and Discovery in Relational Databases

TL;DR

This paper analyzes the computational complexity of detecting and discovering various multi-column dependencies in relational databases, establishing new complexity classifications and equivalences for these problems.

Contribution

It provides a comprehensive complexity classification for dependency detection and discovery, including new W[2]-complete and W[3]-complete results, and relates these problems to hypergraph and Boolean formula enumeration.

Findings

Detection of UCCs and FDs is W[2]-complete.

Discovery of minimal UCCs relates to hypergraph hitting sets.

Detection of INDs is W[3]-complete.

Abstract

Multi-column dependencies in relational databases come associated with two different computational tasks. The detection problem is to decide whether a dependency of a certain type and size holds in a given database, the discovery problem asks to enumerate all valid dependencies of that type. We settle the complexity of both of these problems for unique column combinations (UCCs), functional dependencies (FDs), and inclusion dependencies (INDs). We show that the detection of UCCs and FDs is W[2]-complete when parameterized by the solution size. The discovery of inclusion-wise minimal UCCs is proven to be equivalent under parsimonious reductions to the transversal hypergraph problem of enumerating the minimal hitting sets of a hypergraph. The discovery of FDs is equivalent to the simultaneous enumeration of the hitting sets of multiple input hypergraphs. We further identify the detection of INDs as one of the first natural W[3]-complete problems. The discovery of maximal INDs is shown to be equivalent to enumerating the maximal satisfying assignments of antimonotone, 3-normalized Boolean formulas.