Targeted Least Cardinality Candidate Key for Relational Databases

TL;DR

This paper introduces the TCAND problem, a generalization of the NP-hard least candidate key problem in databases, analyzes its complexity, proposes approximation algorithms, and reveals its relation to the set cover problem and its variants.

Contribution

It formulates the TCAND problem, provides an integer programming approach, analyzes approximation bounds, and establishes its connection to complex set cover variants.

Findings

Approximation algorithms for TCAND have limited improvement potential.

The generalized TCAND problem relates to the red-blue set cover, which is hard to approximate.

The problem's complexity exceeds that of standard set cover.

Abstract

Functional dependencies (FDs) are a central theme in databases, playing a major role in the design of database schemas and the optimization of queries. In this work, we introduce the {\it targeted least cardinality candidate key problem} (TCAND). This problem is defined over a set of functional dependencies $F$ and a target variable set $T \subseteq V$, and it aims to find the smallest set $X \subseteq V$ such that the FD $X \to T$ can be derived from $F$. The TCAND problem generalizes the well-known NP-hard problem of finding the least cardinality candidate key~\cite{lucchesi1978candidate}, which has been previously demonstrated to be at least as difficult as the set cover problem. We present an integer programming (IP) formulation for the TCAND problem, analogous to a layered set cover problem. We analyze its linear programming (LP) relaxation from two perspectives: we propose two approximation algorithms and investigate the integrality gap. Our findings indicate that the approximation upper bounds for our algorithms are not significantly improvable through LP rounding, a notable distinction from the standard set cover problem. Additionally, we discover that a generalization of the TCAND problem is equivalent to a variant of the set cover problem, named red-blue set cover~\cite{carr1999red}, which cannot be approximated within a sub-polynomial factor in polynomial time under plausible conjectures~\cite{chlamtavc2023approximating}. Despite the extensive history surrounding the issue of identifying the least cardinality candidate key, our research contributes new theoretical insights, novel algorithms, and demonstrates that the general TCAND problem poses complexities beyond those encountered in the set cover problem.