The Complexity of Finding Missing Answer Repairs

2026-01-11

arXiv.org

AI Summary

This study investigates the existence and computational complexity of missing query answers in database repair. Focusing on unions of conjunctive queries with negated atoms, it delineates the boundary of polynomial-time solvability for repair problems under combined complexity when weak projection and selection are permitted, and demonstrates that the coexistence of projection and join induces NP-hardness as well as set-cover-type approximation hardness. The main contributions include establishing the OptP[log n]-completeness of computing the size of a minimal repair and an EXP lower bound in the recursive setting; proving that semi-positive Datalog programs can identify minimal repairs in polynomial time under data complexity; and presenting a repair construction algorithm requiring only O(n²) calls to an NP oracle.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We investigate the problem of identifying database repairs for missing tuples in query answers. We show that when the query is part of the input - the combined complexity setting - determining whether or not a repair exists is polynomial-time is equivalent to the satisfiability problem for classes of queries admitting a weak form of projection and selection. We then identify the sub-classes of unions of conjunctive queries with negated atoms, defined by the relational algebra operations permitted to appear in the query, for which the minimal repair problem can be solved in polynomial time. In contrast, we show that the problem is NP-hard, as well as set cover-hard to approximate via strict reductions, whenever both projection and join are permitted in the input query. Additionally, we show that finding the size of a minimal repair for unions of conjunctive queries (with negated atoms permitted) is OptP[log(n)]-complete, while computing a minimal repair is possible with O($n^2$) queries to an NP oracle. With recursion permitted, the combined complexity of all of these variants increases significantly, with an EXP lower bound. However, from the data complexity perspective, we show that minimal repairs can be identified in polynomial time for all queries expressible as semi-positive datalog programs.

Problem

• missing answer repairs
• database repair
• conjunctive queries
• data complexity
• combined complexity

Innovation

• database repair
• missing answers
• conjunctive queries with negation
• combined complexity
• data complexity