关系数据库的目标最小卡片性候选密钥

Vasileios Nakos, Hung Q. Ngo, Charalampos E. Tsourakakis
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引用次数: 0

摘要

功能依赖(FDs)是数据库的核心主题,在数据库模式设计和查询优化中发挥着重要作用。在这项工作中,我们引入了{it targeted least cardinality candidate keyproblem}(TCAND)。这个问题是在一组函数依赖$F$和一个目标变量集$T \subseteq V$上定义的,它的目的是找到最小的集$X \subseteq V$,使得FD $X \to T$ 可以从$F$中导出。TCAND问题概括了众所周知的NP-hard问题--寻找最小卡最小度候选密钥(leastcardinality candidate key~cite{lucchesi1978candidate} )。我们提出了 TCAND 问题的整数编程(IP)公式,类似于分层集合覆盖问题。我们从两个角度分析了它的线性规划(LP)松弛:我们提出了两种近似算法,并研究了积分差距。我们的研究结果表明,我们算法的近似上限并不能通过 LP 舍入得到明显改善,这是与标准集合覆盖问题的显著区别。此外,我们还发现,TCAND 问题的一个泛化等价于集合覆盖问题的一个变种,被命名为红蓝集合覆盖~\cite{carr1999red},在可信猜想~\cite{chlamtavc2023approximating}下,它无法在多项式时间内以亚多项式因子逼近。尽管围绕识别最小卡片数候选密钥的问题已有大量研究,但我们的研究贡献了新的理论见解、新的算法,并证明了一般 TCAND 问题所带来的复杂性超出了集合覆盖问题所遇到的复杂性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Targeted Least Cardinality Candidate Key for Relational Databases
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.
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