Query evaluation over probabilistic databases is notoriously intractable -- not only in combined complexity, but often in data complexity as well. This motivates the study of approximation algorithms, and particularly of combined FPRASes, with runtime polynomial in both the query and instance size. In this paper, we focus on tuple-independent probabilistic databases over binary signatures, i.e., probabilistic graphs, and study when we can devise combined FPRASes for probabilistic query evaluation. We settle the complexity of this problem for a variety of query and instance classes, by proving both approximability results and (conditional) inapproximability results together with (unconditional) DNNF provenance circuit size lower bounds. This allows us to deduce many corollaries of possible independent interest. For example, we show how the results of Arenas et al. [ACJR21a] on counting fixed-length strings accepted by an NFA imply the existence of an FPRAS for the two-terminal network reliability problem on directed acyclic graphs, a question asked by Zenklusen and Laumanns [ZL11]. We also show that one cannot extend a recent result of van Bremen and Meel [vBM23] giving a combined FPRAS for self-join-free conjunctive queries of bounded hypertree width on probabilistic databases: neither the bounded-hypertree-width condition nor the self-join-freeness hypothesis can be relaxed. We last show how our methods can give insights on the evaluation and approximability of regular path queries (RPQs) on probabilistic graphs in the data complexity perspective, showing in particular that some of them are (conditionally) inapproximable.

翻译：在概率数据库上进行查询求值众所周知是难解的——不仅在组合复杂性上如此，在数据复杂性上也常常如此。这促使了对近似算法的研究，特别是对组合FPRAS（全多项式时间随机近似方案）的研究，其运行时间在查询规模和实例规模上均为多项式。本文聚焦于二元签名上的元组独立概率数据库，即概率图，并研究何时能为概率查询求值设计组合FPRAS。我们通过证明近似性结果和（条件性）不可近似性结果，以及（无条件性）DNNF溯源电路规模下界，解决了多种查询和实例类别的该问题复杂性。这使我们能够推导出许多可能具有独立意义的推论。例如，我们展示了Arenas等人[ACJR21a]关于计算由NFA接受的固定长度字符串数量的结果如何蕴含了在有向无环图上两终端网络可靠性问题存在FPRAS，这是Zenklusen和Laumanns[ZL11]提出的问题。我们还表明，无法扩展van Bremen和Meel[vBM23]最近给出的结果，该结果为概率数据库上具有有界超树宽度的无自连接合取查询提供了组合FPRAS：既不能放宽有界超树宽度条件，也不能放松无自连接假设。最后，我们展示了我们的方法如何从数据复杂性视角为概率图上正则路径查询（RPQ）的求值和近似性提供见解，特别指出其中一些查询是（条件性）不可近似的。