Optimal Repairs for Unary Functional Dependencies: Resolving the Case of Updates

2026-08-15

AI Summary

This study addresses the long-standing open problem of classifying the computational complexity of optimal update repairs for unary functional dependency violations. By integrating computational complexity analysis with database consistency repair theory, this work establishes a comprehensive classification through rigorous reduction proofs. The core contribution is the identification of a complexity dichotomy: the problem is either solvable in polynomial time or NP-hard, with no intermediate complexity classes. This result definitively resolves theoretical challenges regarding optimal repair complexity within the field and bridges critical gaps in existing literature. Furthermore, it provides a complete theoretical foundation and precise boundary conditions to guide the design of algorithms for data consistency repair.

Abstract

If a table violates its required set of functional dependencies (FDs), what is the minimum number of cell changes needed to restore consistency? This fundamental problem, known as finding an optimal update repair (U-repair), is known to admit polynomial-time algorithms only for a small number of specific FD sets. Whether additional tractable cases exist has remained open. The only established hardness result for this problem is due to Kolahi and Lakshmanan (2009); subsequent attempts to prove hardness for additional cases have failed, leaving these cases unresolved. In this work, we make substantial progress on this open problem by completely resolving the case of unary FDs, in which every FD has a single attribute on its left-hand side. We show that every set of unary FDs either falls into one of the previously known tractable classes or makes the problem of finding an optimal U-repair NP-hard.

Problem

• Optimal Update Repair
• Unary Functional Dependencies
• Computational Complexity
• Database Consistency
• NP-hardness

Innovation

• Optimal U-repair
• Unary Functional Dependencies
• Computational Complexity
• NP-hardness Dichotomy