Regular path queries (RPQs) are an essential component of graph query languages. Such queries consider a regular expression r and a directed edge-labeled graph G and search for paths in G for which the sequence of labels is in the language of r. In order to avoid having to consider infinitely many paths, some database engines restrict such paths to be trails, that is, they only consider paths without repeated edges. In this paper we consider the evaluation problem for RPQs under trail semantics, in the case where the expression is fixed. We show that, in this setting, there exists a trichotomy. More precisely, the complexity of RPQ evaluation divides the regular languages into the finite languages, the class Ttract (for which the problem is tractable), and the rest. Interestingly, the tractable class in the trichotomy is larger than for the trichotomy for simple paths, discovered by Bagan, Bonifati, and Groz [JCSS 2020]. In addition to this trichotomy result, we also study characterizations of the tractable class, its expressivity, the recognition problem, closure properties, and show how the decision problem can be extended to the enumeration problem, which is relevant to practice.

翻译：正则路径查询是图查询语言的重要组成部分。此类查询考虑正则表达式r和带标记边的有向图G，并在G中寻找标签序列属于r所定义语言的路径。为避免考虑无限多条路径，某些数据库引擎将路径限制为踪迹，即仅考虑不含重复边的路径。本文研究了在表达式固定的情况下基于踪迹语义的正则路径查询评估问题。我们证明在此设定下存在三分法：正则语言可分为有限语言、类Ttract（该问题在此类中可解）及其他语言三类。值得注意的是，三分法中可解类比Bagan、Bonifati和Groz [JCSS 2020]发现的简单路径三分法中的可解类更大。除三分法结果外，我们还研究了可解类的特征刻画、表达能力、识别问题、封闭性质，并展示了如何将判定问题扩展至具有实际应用价值的枚举问题。