A key task in the context of consistent query answering is to count the number of repairs that entail the query, with the ultimate goal being a precise data complexity classification. This has been achieved in the case of primary keys and self-join-free conjunctive queries (CQs) via an FP/#P-complete dichotomy. We lift this result to the more general case of functional dependencies (FDs). Another important task in this context is whenever the counting problem in question is intractable, to classify it as approximable, i.e., the target value can be efficiently approximated with error guarantees via a fully polynomial-time randomized approximation scheme (FPRAS), or as inapproximable. Although for primary keys and CQs (even with self-joins) the problem is always approximable, we prove that this is not the case for FDs. We show, however, that the class of FDs with a left-hand side chain forms an island of approximability. We see these results, apart from being interesting in their own right, as crucial steps towards a complete classification of approximate counting of repairs in the case of FDs and self-join-free CQs.

翻译：一致查询回答中的一个关键任务是计算蕴含查询的修复数量，其最终目标是实现精确的数据复杂度分类。在主键和无自连接合取查询（CQs）的情况下，已通过FP/#P完全二分性实现了这一目标。我们将这一结果推广到更一般的函数依赖（FDs）情形。另一项重要任务是当计数问题难以处理时，将其分类为可近似问题（即目标值可通过全多项式随机近似方案（FPRAS）在误差保证下高效近似）或不可近似问题。尽管对于主键和CQs（甚至包含自连接的情况），该问题总是可近似的，但我们证明对于FDs并非如此。然而，我们表明具有左部链的FDs类构成了可近似性的孤岛。我们视这些结果（除了其本身的意义外）为在FDs和无自连接CQs条件下完成修复近似计数完全分类的关键步骤。