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.

翻译：我们研究了在查询结果中识别缺失元组的数据库修复问题。我们证明，当查询作为输入的一部分（即组合复杂性设定）时，判断修复是否存在是多项式时间可解的，等价于允许弱形式投影和选择操作的查询类别的可满足性问题。随后，我们确定了允许在查询中出现特定关系代数操作的、带否定原子的合取查询并集的子类，对于这些子类，最小修复问题可在多项式时间内求解。相反，我们证明当输入查询同时允许投影和连接操作时，该问题不仅是NP难的，而且通过严格归约近似求解也是集合覆盖难的。此外，我们发现对于（允许带否定原子的）合取查询并集，计算最小修复的规模是OptP[log(n)]-完全的，而计算具体的最小修复可通过O($n^2$)次NP预言机查询实现。当允许递归时，所有这些变体的组合复杂性显著增加，存在EXP下界。然而，从数据复杂性的角度，我们证明对于所有可表达为半正数datalog程序的查询，最小修复均可在多项式时间内识别。