A Trichotomy for Regular Simple Path Queries on Graphs

TL;DR

This paper classifies the complexity of evaluating regular simple path queries on graphs, identifying the boundary between tractable and intractable cases, and providing a detailed complexity trichotomy based on the regular language involved.

Contribution

It establishes a complete classification of regular languages for RSPQ evaluation complexity, revealing a trichotomy and characterizing the tractability frontier.

Findings

Evaluation complexity depends on the regular language: AC0, NL-complete, or NP-complete.

The tractability frontier is characterized by a simple regular expression fragment.

Deciding tractability of finding a regular simple path is NL-complete for DFAs and PSPACE-complete for NFAs or regexes.

Abstract

Regular path queries (RPQs) select nodes connected by some path in a graph. The edge labels of such a path have to form a word that matches a given regular expression. We investigate the evaluation of RPQs with an additional constraint that prevents multiple traversals of the same nodes. Those regular simple path queries (RSPQs) find several applications in practice, yet they quickly become intractable, even for basic languages such as (aa)* or a*ba*. In this paper, we establish a comprehensive classification of regular languages with respect to the complexity of the corresponding regular simple path query problem. More precisely, we identify the fragment that is maximal in the following sense: regular simple path queries can be evaluated in polynomial time for every regular language L that belongs to this fragment and evaluation is NP-complete for languages outside this fragment. We thus fully characterize the frontier between tractability and intractability for RSPQs, and we refine our results to show the following trichotomy: Evaluations of RSPQs is either AC0, NL-complete or NP-complete in data complexity, depending on the regular language L. The fragment identified also admits a simple characterization in terms of regular expressions. Finally, we also discuss the complexity of the following decision problem: decide, given a language L, whether finding a regular simple path for L is tractable. We consider several alternative representations of L: DFAs, NFAs or regular expressions, and prove that this problem is NL-complete for the first representation and PSPACE-complete for the other two. As a conclusion we extend our results from edge-labeled graphs to vertex-labeled graphs and vertex-edge labeled graphs.