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.

Modern graph database query languages such as GQL, SQL/PGQ, and their academic predecessor G-Core promote paths to first-class citizens in the sense that their pattern matching facility can return paths , as opposed to only nodes and edges. This is challenging for database engines, since graphs can have a large number of paths between a given node pair, which can cause huge intermediate results in query evaluation. We introduce the concept of path multiset representations (PMRs) , which can represent multisets of paths exponentially succinctly and therefore bring significant advantages for representing intermediate results. We give a detailed theoretical analysis that shows that they are especially well-suited for representing results of regular path queries and extensions thereof involving counting, random sampling, and unions. Our experiments show that they drastically improve scalability for regular path query evaluation, with speedups of several orders of magnitude.

We study the data complexity of regular trail and simple path queries on undirected graphs. Using techniques from structural graph theory, ranging from the graph minor theorem to group-labeled graphs, we are able to identify several tractable and intractable subclasses of the regular languages. In particular, we establish that trail evaluation for simple chain regular expressions, which are common in practice, is tractable, whereas simple path evaluation is tractable for a large subclass. The problem of fully classifying all regular languages is quite non-trivial, even on undirected graphs, since it subsumes an intriguing problem that has been open for 30 years.

Testing containment of queries is a fundamental reasoning task in knowledge representation. We study here the containment problem for Conjunctive Regular Path Queries (CRPQs), a navigational query language extensively used in ontology and graph database querying. While it is known that containment of CRPQs is EXPSPACE-complete in general, we focus here on severely restricted fragments, which are known to be highly relevant in practice according to several recent studies. We obtain a detailed overview of the complexity of the containment problem, depending on the features used in the regular expressions of the queries, with completeness results for NP, Pi2p, PSPACE or EXPSPACE.

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 T_tract (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 et al. [Bagan et al., 2013]. 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.

Regular path queries (RPQs) are a central component of graph databases. We investigate decision and enumeration problems concerning the evaluation of RPQs under several semantics that have recently been considered: arbitrary paths, shortest paths, paths without node repetitions (simple paths), and paths without edge repetitions (trails). Whereas arbitrary and shortest paths can be dealt with efficiently, simple paths and trails become computationally difficult already for very small RPQs. We study RPQ evaluation for simple paths and trails from a parameterized complexity perspective and define a class of simple transitive expressions that is prominent in practice and for which we can prove dichotomies for the evaluation problem. We observe that, even though simple path and trail semantics are intractable for RPQs in general, they are feasible for the vast majority of RPQs that are used in practice. At the heart of this study is a result of independent interest: the two disjoint paths problem in directed graphs is W[1]-hard if parameterized by the length of one of the two paths.

Evaluation of regular path queries (RPQs) is a central problem in graph databases. We investigate the corresponding enumeration problem, that is, given a graph and an RPQ, enumerate all paths in the graph that match the RPQ. We consider several versions of this problem, corresponding to different semantics of RPQs that have recently been considered: arbitrary paths, shortest paths, simple paths, and trails. Whereas arbitrary and shortest paths can be enumerated in polynomial delay, the situation is much more intricate for simple paths and trails. For instance, already the question if a given graph contains a simple path or trail of a certain length has cases with highly non-trivial solutions and cases that are long-standing open problems. In this setting, we study RPQ evaluation from a parameterized complexity perspective. We define a class of simple transitive expressions that is prominent in practice and for which we can prove two dichotomy-like results: one for simple paths and one for trails paths. We observe that, even though simple path semantics and trail semantics are intractable for RPQs in general, they are feasible for the vast majority of the kinds of RPQs that users use in practice. At the heart of this study is a result of independent interest on the parameterized complexity of finding disjoint paths in graphs: the two disjoint paths problem is W[1]-hard if parameterized by the length of one of the two paths.