We investigate the complexity of computing an optimal repair of an inconsistent database, in the case where integrity constraints are Functional Dependencies (FDs). We focus on two types of repairs: an optimal subset repair (optimal S-repair), which is obtained by a minimum number of tuple deletions, and an optimal update repair (optimal U-repair), which is obtained by a minimum number of value (cell) updates. For computing an optimal S-repair, we present a polynomial-time algorithm that succeeds on certain sets of FDs and fails on others. We prove the following about the algorithm. When it succeeds, it can also incorporate weighted tuples and duplicate tuples. When it fails, the problem is NP-hard and, in fact, APX-complete (hence, cannot be approximated better than some constant). Thus, we establish a dichotomy in the complexity of computing an optimal S-repair. We present general analysis techniques for the complexity of computing an optimal U-repair, some based on the dichotomy for S-repairs. We also draw a connection to a past dichotomy in the complexity of finding a “most probable database” that satisfies a set of FDs with a single attribute on the left-hand side; the case of general FDs was left open, and we show how our dichotomy provides the missing generalization and thereby settles the open problem.

中文翻译：

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计算功能依赖的最优修复

在完整性约束是功能依赖关系 (FD) 的情况下，我们研究了计算不一致数据库的最佳修复的复杂性。我们关注两种类型的修复：最优子集修复（optimal S-repair），通过最小数量的元组删除获得，以及最优更新修复（optimal U-repair），通过最小数量获得值（单元格）更新。为了计算最优的 S 修复，我们提出了一种多项式时间算法，该算法在某些 FD 集上成功而在其他 FD 集上失败。我们证明以下关于算法的内容。当它成功时，它还可以合并加权元组和重复元组。当它失败时，问题是 NP-hard，实际上是 APX-complete（因此，不能比某个常数更好地近似）。因此，我们在计算最优 S 修复的复杂性中建立了二分法。我们提出了计算最优 U 修复的复杂性的一般分析技术，其中一些基于 S 修复的二分法。我们还与过去的二分法联系起来，即找到一个“最可能的数据库”，该数据库满足左侧具有单个属性的一组 FD；一般FD的情况是开放的，我们展示了我们的二分法如何提供缺失的概括，从而解决开放问题。