Abstract A fundamental query in labeled graphs is to determine if there exists a path between a given source and target vertices, such that the path satisfies a given label constraint. One of the powerful forms of specifying label constraints is through regular expressions, and the resulting problem of reachability queries under regular simple paths (RSP) form the core of many practical graph query languages such as SPARQL from W3C, Cypher of Neo4J, Oracle’s PGQL and LDBC’s G-CORE. Despite its importance, since it is known that answering RSP queries is NP-Hard, there are no scalable and practical solutions for answering reachability with full-range of regular expressions as constraints. In this paper, we circumvent this computational bottleneck by designing a random-walk based sampling algorithm called ARRIVAL, which is backed by theoretical guarantees on its expected quality. Extensive experiments on billion-sized real graph datasets with thousands of labels show that ARRIVAL to be 100 times faster than baseline strategies with an average accuracy of 95%.